Continuous time random walks and queues: explicit forms and approximations of the conditional law

with respect to their local times ^{∗}

Giovanna Nappo^{†}  Barbara Torti ^{‡}

Abstract

A filtering problem is considered in the case when the state process is a continuous time random walk and the observation process is its local time. An explicit construction of the filter is given. This construction is then applied to a suitable approximation of a Brownian motion and to a rescaled M/M/1 queueing model. In both these cases a convergence result for the respective filters is given. The case of a queueing model when the observation is the idle time is also considered.

MSC: 60K20; 60J55; 60F99; 60G35; 93E11; 60G55

Keywords: Queues; Idle time; Nonlinear filtering; Point processes; Weak convergence; Reflection principle

Contents

1 Introduction 2

2 The model 5

3 Scaling 8

4 The interpolating Brownian motion model 10

5 The M/M/1 queueing model 13 5.1 The symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 The M/M/1 queueing model: observing the idle time process 31

∗Partially supported by MURST project ”Processi Stocastici, Modelli Stocastici ed Applicazioni”

†Dipartimento di Matematica - Università di Roma ”La Sapienza”, Italy ‡Dipartimento di Matematica - Università di Roma ”Tor Vergata”, Italy

1 Introduction

The kind of problems we are interested in arises from the following situation. Suppose that in a queue we can observe, up to time t, whether the queue is busy or idle, but we cannot observe the size of the queue, so that the observation process is the total time the queue has spent in 0, i.e. the so called idle time (see Prabhu [10]). Then the problem is to evaluate the size of the queue at time t, given this information, i.e. to compute the conditional law (or the filter) of the queue given the observation process up to time t. In the setup of heavy traffic limit, the rescaled queue converges to a reflected Brownian motion and the observation process converges to its local time. Then the limit model can be constructed as (W_{t} + Λ_{t}, Λ_{t}), where W_{t} is a Brownian motion and Λ_{t} is the local time in the sense of Skorohod definition, that is

Λ_{t} = ℓ_{t}(W),

where

ℓ : D_{ℝ}[0, ∞) → D_{ℝ}[0, ∞), x → ℓ(x), such that ℓ_{t}(x) = − inf_{s≤t} x(s) ∧ 0.  (1)

In the limit model the corresponding filtering problem is the computation of the conditional law of a reflected Brownian motion W_{t} + Λ_{t} when the observation process is its local time Λ_{t}, i.e. the computation of the filter E[g(W_{t} + Λ_{t})/F_{t}^{Λ}], for g in a sufficiently large class of functions. A first problem is to find the exact expression for the filters, both for the limit model and for the rescaled queue model. A second problem concerns the convergence of the latter to E[g(W_{t} + Λ_{t})/F_{t}^{Λ}]. The filter of the limit Brownian motion model is derived in G. Nappo, B. Torti [9] (Sections 4 and 6), where it is obtained by means of a suitable sequence of processes Λ^{n} approximating the observation process Λ. Each process Λ^{n} is proportional to a counting process, and therefore the nonlinear filtering techniques for counting processes are used. The filter can also be derived by means of the Azéma martingale, and this derivation is shortly discussed in [9]. For sake of completeness we recall its explicit expression.

Theorem 1.1. Let W_{t} be a Brownian motion with diffusion coefficient a^{2} and drift c ∈ ℝ and let Λ_{t} be its local time. Let g be a bounded measurable function. Denote by

Π(s, l; g) = ∫_{0}^{∞} g (−l + y\sqrt{s}) y exp (−\frac{y^{2}}{2} ) dy s ≥ 0,  (2)

Π_{a^{2},c}(s, l; g) = \frac{Π (a^{2}s, l; g(·) exp(\frac{c}{a^{2}} ·))}{Π (a^{2}s, l; exp(\frac{c}{a^{2}} ·))}  (3)

Π̂_{a^{2},c}(s; g) = Π_{a^{2},c}(s, 0; g)  (4)

Then

π_{t}(g) = E[g(W_{t})/F_{t}^{Λ}] = Π_{a^{2},c}(ζ_{t}, Λ_{t}; g),  (5)

and

π̂_{t}(g) = E[g(W_{t} + Λ_{t})/F_{t}^{Λ}] = Π̂_{a^{2},c}(ζ_{t}; g)  (6)

where F_{t}^{Λ} is the history generated by Λ_{u} up to time t, ζ_{t} is the elapsed time from the last visit to 0 for the process W_{t} + Λ_{t}, i.e.

ζ_{t} = γ_{t}^{0}(W + Λ) = γ_{t}(Λ),  (7)

with

γ_{t}^{0}(x) = t − sup{s < t : x_{s} = 0}.  (8)
γ_{t}(x) = t − sup{s < t : x_{s} < x_{t}}.  (9)

Note that Π_{1,0}(s, l; g) = Π(s, l; g) = E[g(−l+W_{s}^{∗})/Λ_{s}^{∗} = 0], where W^{∗} is any standard Brownian motion and Λ^{∗} is its local time. In this paper we start by considering a somehow simplified version of the motivating problem: we consider a continuous time random walk Y_{t} and its conditional law w.r.t. its local time L_{u} = ℓ_{u}(Y) up to time t. Indeed in several cases a queueing model can be represented as the reflection of a continuous time random walk. It turns out that the filter of Y_{t} w.r.t. L_{u} up to time t can be expressed as a probability measure depending deterministically on L_{t} and γ_{t}(L), where γ_{t} is defined in (9). We derive also a more explicit expression for the filter under the assumption that the process Y_{t} can be decomposed as Y_{t} = V_{Z_{t}}, where Z_{t} is a renewal process, V_{k} is a discrete time random walk, with Z_{t} and V_{k} mutually independent. Similar results hold also for rescaled random walks.

We are interested in the situation when a sequence X_{t}^{n} of rescaled random walks converges to a Brownian motion W_{t} in D_{ℝ}[0, +∞). Then a continuous map argument applies to show that (X_{t}^{n}, L_{t}^{n}) = (X_{t}^{n}, ℓ_{t}(X^{n})) converges to (W_{t}, Λ_{t}) = (W_{t}, ℓ_{t}(W)), and analogously the systems with the reflected random walk (X_{t}^{n} + L_{t}^{n}, L_{t}^{n}) converge to the system with the reflected Brownian motion (W_{t} + Λ_{t}, Λ_{t}) (see Section 5 for the details). In analogy with the second equality of (7) we denote

ξ_{t}^{n} = γ_{t}(L^{n}) = t − sup{u ≤ t such that L_{u}^{n} < L_{t}^{n}}.  (10)

the elapsed time from last jump time of L^{n}. Then for random walk systems the corresponding filter is

π_{t}^{n}(g) = E[g(X_{t}^{n})/F_{t}^{L^{n}}] = Σ^{n}(ξ_{t}^{n}, L_{t}^{n}; g)  (11)

with Σ^{n}(s, l) defined in (23), while for the reflected systems is

π̂_{t}^{n}(g) = E[g(X_{t}^{n} + L_{t}^{n})/F_{t}^{L^{n}}] = Σ̂^{n}(ξ_{t}^{n}; g)  (12)

with Σ̂^{n}(s) = Σ^{n}(s, 0). The problem whether π_{t}^{n} = Σ^{n}(γ_{t}(L^{n}), L_{t}^{n}) converge weakly to the filter π_{t} = Π_{a^{2},c}(ζ_{t}, Λ_{t}) of the limit is strictly related to the convergence of the filters π̂_{t}^{n} = Σ̂^{n}(γ_{t}(L^{n})) for the reflected random walks to π̂_{t} = Π̂_{a^{2},c}(ζ_{t}). Depending on the model we privilege one or the other problem. On the other hand, from a computational point of view, it is quite difficult to use the exact expression of Σ̂^{n}(s) to compute the filter π̂_{t}^{n}. Then it is also interesting to find a good approximation of the filter of the discrete system, depending on the actually observed process L^{n}, so that it can be used in applications. For the reflected random walk system, a natural choice in order to give a manageable approximation of the filter is to use the limit functional Π̂_{a^{2},c}(s), evaluated at s = ξ_{t}^{n}.

These problems have been studied in more general situations by many authors, among which we recall in particular Bhatt et al. in [1] and Goggin ([6], [7]). Most of these results concern diffusive models and do not apply to our case. Moreover, usually the applications concern the problem to approximate a given signal/observation process with a suitably chosen sequence of signal/observation processes so that the corresponding sequence of filters converges to the the filter of the original process. We start from a different point of view: the sequence of processes is given and the problem is to show the convergence of the filters to the filter of the state/observation limit, in the sense specified above.

The problem of weak convergence of the sequence of filters is not a trivial problem as even strong convergence of random variables does not imply convergence of the conditional laws. This is clearly explained by the following simple and illuminating example (see E. Goggin [6]). Let ξ be a real random variable, and (ξ_{n}, η_{n}) = (ξ, ξ/n). Then (ξ_{n}, η_{n}) converges strongly to (ξ, η), with η = 0. Nevertheless, for any measurable function g, E(g(ξ_{n})/η_{n}) = g(ξ), so that the conditional law of ξ_{n} given η_{n} is the measure concentrated in ξ(ω), while E(g(ξ)/η) = E(g(ξ)), so that the conditional law of ξ given η coincides with the (deterministic) law P ∘ ξ^{−1} of ξ. In this example, although the sequence of the conditional laws of ξ_{n} given η_{n} does not converge to the conditional law of the limit, it is a constant sequence and therefore is a converging sequence. This is not surprising indeed in the light of the next general result, which is a slight generalization of a result of E. Goggin [6] (proof of Theorem 2.1, Step 1): one has only to replace the sequence of σ−algebras used in [6] with a general sequence.

Lemma 1.2. Let R_{n} be a sequence of random variables with values in a Polish space, let H^{n} be a sequence of σ−algebras, let α^{n} be a regular version of the conditional distribution of R_{n} given H^{n}. If {R_{n}, n ∈ ℕ} is tight, then {α^{n}, n ∈ ℕ} is tight.

Then, as far as weak convergence is concerned, for the systems converging to a Brownian motion, the main problem is to check whether the limit points of the sequence of filters are all equal to the filter of a standard Brownian motion w.r.t. its local time.

The first model we consider is a non-Markovian queueing model arising when a Brownian motion W is approximated by a sequence of continuous time random walks W^{n}, obtained with a suitable interpolation procedure. The approximation scheme we propose for W follows some of the ideas used in [7] to study a filter approximation problem in diffusive models, and is related with the approximation scheme used in [9]. In this particular case we get a strong convergence result for the approximating filter. The second model we consider is the case when the renewal process of the random walk is a Poisson process, so that the reflected random walk is an M/M/1 queue. The main results are weak convergence of the corresponding filters (see Theorem 5.1) and approximation in L^{p}(Ω × [0, T])-norm (see Theorem 5.2) and are based on the weak convergence of (ξ_{t}^{n}, L_{t}^{n}) to (ζ_{t}, Λ_{t}) (Proposition 5.3) and on the convergence of Σ̂^{n}(·) to Π̂_{a^{2},c}(·), in the sense that

lim_{n→∞} Σ̂^{n}(s_{n}; g) = Π̂_{2λ,c}(s; g),  (13)

for any g in a convergence determining class, whenever s_{n} converges to s, with s > 0. We first consider the case when the random walk is symmetric and so the drift of the limit Brownian motion is zero, and, under suitable conditions, we prove the above key convergence result (13) by using the reflection principle (see Section 5.1). Then we consider the general case in Section 5.2, where we prove (13) by reformulating the problem in terms of the symmetric random walk by using a suitable change of measure. It is important to note that when we deal with a queue Q_{t}^{n} = X_{t}^{n} + L_{t}^{n}, i.e. with a reflected random walk, the previous results concern the filter w.r.t. the filtration

G_{t}^{n} = F_{t}^{L^{n}}

generated by the local time of the random walk, while the motivating problem concerns the filter of the queue w.r.t. the filtration

H_{t}^{n} = F_{t^{+}}^{C^{n}}

generated by the idle time C_{t}^{n}, i.e. the total time spent in 0 up to t. These two problems are strictly related, for instance (Q_{t}^{n}, C_{t}^{n}), as well as (Q_{t}^{n}, L_{t}^{n}), converges weakly to the reflected system (W_{t}+Λ_{t}, Λ_{t}). This property, among others, allows to extend the previous convergence and approximation results to this situation (see Theorem 6.4 and Theorem 6.6 in the last section).

2 The model

Fix a probability space (Ω, F, P) and consider on it a sequence {(T_{j}, U_{j}), j ≥ 1}, satisfying the assumption

H0 The ℝ^{+} × {+1, −1}−valued random variables (T_{j}, U_{j}), for j ≥ 1, are identically distributed and mutually independent.

Put

τ_{0} = 0, τ_{k} = ∑_{j=1}^{k} T_{j} for k ≥ 1,

consider the renewal process

Z_{t} = ∑_{j=1}^{∞} 𝕀 (τ_{j} ≤ t)

and the random walk {V_{j}, j ≥ 0} defined by

V_{0} = 0, V_{j} = V_{j−1} + U_{j}, j ≥ 1.

Finally, consider the continuous time random walk

Y_{t} = V_{Z_{t}} = ∑_{j=1}^{∞} U_{j}𝕀 (τ_{j} ≤ t) = ∑_{j≤Z_{t}} U_{j}.  (14)

The solution of the Skorohod problem ^{1} for the process Y_{t} is given by the pair (Y_{t} +L_{t}, L_{t}) where L_{t} is the local time at level 0 for the process Y_{t}, that is

L_{t} = ℓ_{t}(Y),

where ℓ is defined in (1). It is easy to see that, for our model, L_{t} admits the representation

L_{t} = ∑_{j=1}^{∞} 𝕀 (σ_{j} ≤ t) ,  (15)

and that the sequence of its jump times {σ_{j}, j ≥ 0} is the subsequence of {τ_{j}, j ≥ 0} defined by σ_{0} = 0 and

σ_{j} = inf {τ_{k} s.t. Y_{τ_{k}} ≤ −j} = inf {t > 0 s.t. Y_{t} ≤ −j} for j ≥ 1.

1For any x ∈ D_{ℝ}[0, ∞), with x(0) ≥ 0 the pair (z, v) = (x + ℓ(x), ℓ(x)) is the unique solution of the Skorohod problem, i.e. is the unique pair of functions satisfying z(t) = x(t) + v(t), and such that z(t) ≥ 0, for all t ≥ 0, v(0) = 0, v is nondecreasing and increases only when z(t) = 0.

Set

G_{t} = F_{t}^{L} = σ {L_{s}, s ≤ t} ,

obviously {σ_{j}, j ≥ 0} are stopping times w.r.t. both the histories G_{t} and F_{t}^{Y}.

A first representation for the filter of Y_{t} given G_{t} is stated in the following Proposition.

Proposition 2.1. Assume H0. Then the conditional law of Y_{t} given G_{t} admits the following representation P -a.s.

E[g(Y_{t})/G_{t}] = ∑_{j=0}^{∞} \frac{E[g(−j + Y_{s+σ_{j}} − Y_{σ_{j}})𝕀 (S_{j+1} > s) ]_{s=t−σ_{j}}}{E[𝕀(S_{j+1} > s)]_{s=t−σ_{j}}} 𝕀\{σ_{j} ≤ t < σ_{j+1}\},  (16)

where S_{j+1} = σ_{j+1} − σ_{j}, j ≥ 0.

Proof. The thesis can be proved by a slight modification of the argument in [9], Proposition 3.1.

Remark 2.2. The main ingredient in the proof of the previous Proposition is that condition H0 implies that the process Y_{s+σ_{j}} − Y_{σ_{j}} is independent of G_{σ_{j}} and is equal in law to the process Y_{s}. In its turn the last property implies that (16) admits the representation

E[g(Y_{t})/G_{t}] = ∑_{j=0}^{∞} \frac{E[g(−j + Y_{s}^{∗})𝕀 (σ_{1}^{∗} > s) ]_{s=t−σ_{j}}}{E[𝕀(σ_{1}^{∗} > s)]_{s=t−σ_{j}}} 𝕀\{σ_{j} ≤ t < σ_{j+1}\},  (17)

where Y_{s}^{∗} is another birth and death process, with the same law as Y_{t}, defined by the rule (14) starting from a sequence {(T_{j}^{∗}, U_{j}^{∗}), ≥ 1} satisfying condition H0, and σ_{1}^{∗} is its first visit time to the state -1. Then the problem reduces to the computation of the probability measure defined by

\frac{E[g(−j + Y_{s}^{∗})𝕀 (σ_{1}^{∗} > s) ]}{E[𝕀(σ_{1}^{∗} > s)]} = \frac{E[g(−j + Y_{s})𝕀 (σ_{1} > s) ]}{E[𝕀(σ_{1} > s)]},

which has to be evaluated at j = L_{t} and s = γ_{t}(L).

The previous Remark implies also that the process L_{t} is a renewal process, with inter-arrival times {S_{h}, h ≥ 1}. It is possible to represent them in terms of the sequences {T_{i}, i ≥ 1} and {M_{i}, i ≥ 0}, where the sequence {M_{i}, i ≥ 0} is recursively defined by the rule

\begin{cases}M_{0} = 0 \\ M_{i} = inf \{k ≥ 0 : V_{M_{0}+...+M_{i−1}+k} − V_{M_{0}+...+M_{i−1}} = −1\} . \end{cases}  (18)

Then

σ_{0} = 0, σ_{h} = ∑_{i=1}^{M_{1}+...+M_{h}} T_{i} = τ_{M_{1}+...+M_{h}}, h ≥ 1,

and

S_{h} = σ_{h} − σ_{h−1} = ∑_{i=M_{1}+...+M_{h−1}+1}^{M_{1}+...+M_{h}} T_{i},  h ≥ 1.

Under Condition H0 the sequence {M_{i}, i ≥ 1} is a sequence of i.i.d. random variables, moreover, for a suitable distribution function F and a suitable value of p ∈ (0, 1), the following conditions hold:

H1 the random variables {T_{j}, j ≥ 1} are non-negative, mutually independent, with common distribution function F,

H2 {U_{j}, j ≥ 1} is a sequence of i.i.d. random variables such that

P (U_{j} = 1) = p P (U_{j} = −1) = 1 − p = q, p ∈ (0, 1).

Under the further condition

H3 the sequences {T_{j}, j ≥ 1} and {U_{j}, j ≥ 1} are mutually independent;

clearly also {T_{i}, i ≥ 1} and {M_{i}, i ≥ 1} are mutually independent. Next result provides a more explicit expression for the terms of the sum in (16).

Proposition 2.3. Assume conditions H1, H2, H3. Then the filter E[g(Y_{t})/G_{t}] admits the representation

E[g(Y_{t})/G_{t}] =
= ∑_{j=0}^{∞} \frac{∑_{k=1}^{∞} E[𝕀 (M_{1} ≥ k) g(−j + V_{k−1})] (F_{k−1}(t − σ_{j}) − F_{k}(t − σ_{j}))}{∑_{m=1}^{∞} P (M_{1} = m) (1 − F_{m}(t − σ_{j}))}𝕀\{σ_{j} ≤ t < σ_{j+1}\} (19)

where F_{k} is the distribution function of τ_{k} i e. F_{k} = F^{∗k}, the k-fold convolution of F.

Proof. As explained in Remark 2.2 we need only to compute E[𝕀(σ_{1} > s)], and E[g(−j + Y_{s})𝕀 (σ_{1} > s) ], for all j ≥ 0. Taking into account the equality σ_{1} = τ_{M_{1}}, and the independence of the sequences {T_{i}, i ≥ 1} and {M_{i}, i ≥ 0}

E[𝕀 (σ_{1} > s) ] = E[𝕀 (τ_{M_{1}} > s) ] = E [∑_{m=1}^{∞} 𝕀 (M_{1} = m) 𝕀 (τ_{m} > s)] = ∑_{m=j}^{∞} P (M_{1} = m)(1 − F_{m}(s)).

Finally, taking into account also (14),

E[g(−j + Y_{s})𝕀 (σ_{1} > s) ] =E [g (−j + V_{Z_{s}}) 𝕀 (τ_{M_{1}} > s)] = E [∑_{m=1}^{∞} 𝕀 (M_{1} = m) g (−j + V_{Z_{s}}) 𝕀 (τ_{m} > s)] =

=E [∑_{m=1}^{∞} ∑_{i=1}^{m} 𝕀 (M_{1} = m) g (−j + V_{i−1}) 𝕀 (τ_{i−1} ≤ s < τ_{i})] =

= ∑_{m=1}^{∞} ∑_{i=1}^{m} E[𝕀 (M_{1} = m) g(−j + V_{i−1})]P (τ_{i−1} ≤ s < τ_{i}) =

= ∑_{i=1}^{∞} ∑_{m=i}^{∞} E[𝕀 (M_{1} = m) g(−j + V_{i−1})](F_{i−1}(s) − F_{i}(s)) =

= ∑_{i=1}^{∞} E[𝕀 (M_{1} ≥ i) g(−j + V_{i−1})](F_{i−1}(s) − F_{i}(s)).

The state space of the process Y_{t} being discrete, the filter E[g(Y_{t})/G_{t}] is determined by its discrete density ν_{t}(x), x ∈ ℤ, i.e. ν_{t}(x) = E[g(Y_{t})/G_{t}] with g(z) = 𝕀_{{x}}(z), z ∈ ℤ. Then the expression of ν_{t}(x) is

ν_{t}(x) = ∑_{j=0}^{∞} \frac{∑_{k=1}^{∞} P (V_{k−1} = x + j, M_{1} ≥ k) (F_{k−1}(t − σ_{j}) − F_{k}(t − σ_{j}))}{∑_{m=1}^{∞} P (M_{1} = m)(1 − F_{m}(t − σ_{j}))}𝕀\{σ_{j} ≤ t < σ_{j+1}\},

where

P (V_{k−1} = a, M_{1} ≥ k) =

= \begin{cases}\frac{a + 1}{k}\binom{k}{\frac{k+a+1}{2}}p^{\frac{k+a−1}{2}} q^{\frac{k−a−1}{2}} & if k + a − 1 is even and |a| ≤ k − 1 \\ 0 & otherwise. \end{cases}  (20)

3 Scaling

Let {X̃^{n}, n ∈ ℕ} be a sequence of continuous time random walks defined in (Ω^{n}, F^{n}, P^{n}). We assume that X̃_{t}^{n} = Ṽ n ; Z̃_{t}^{n}, where Ṽ_{k}^{n} and Z̃_{t}^{n} are defined as in Section 2 starting from a sequence

{(T̃_{j}^{n}, Ũ_{j}^{n}); j ≥ 1}. For each n, we assume that the sequence {(T̃_{j}^{n}, Ũ_{j}^{n}); j ≥ 1} satisfies the assumption H0 stated in Section 2. Consider the deterministic linear time-space scaling

X_{t}^{n} = b_{n}X̃_{a_{n}t}^{n},

where {a_{n}, n ∈ ℕ} and {b_{n}, n ∈ ℕ} are suitable sequences of real numbers. The local time L_{t}^{n} = ℓ_{t}(X^{n}) of the rescaled process X_{t}^{n} can be obtained just applying the same scaling to the process L̃_{t}^{n} = ℓ_{t}(X̃^{n}), i.e.

L_{t}^{n} = b_{n}L̃_{a_{n}t}^{n}.

We are interested in the conditional law of X_{t}^{n} w.r.t. F_{t}^{L^{n}} = F L̃^{n} ; a_{n}t, i.e. the filter

π_{t}^{n}(g) = E^{P^{n}}[g(X_{t}^{n})/F_{t}^{L^{n}}] = E^{P^{n}}[g(b_{n}X̃_{a_{n}t}^{n})/F L̃^{n} ; a_{n}t],

where E^{P^{n}} denotes the expectation w.r.t. P^{n}. For the sake of notational convenience we will denote F_{t}^{L^{n}} as G_{t}^{n} and, when unnecessary, drop the symbol P^{n} in the expectation, so that the filter becomes

π_{t}^{n}(g) = E[g(X_{t}^{n})/G_{t}^{n}].

Following the same lines as in the previous section we get the representation

π_{t}^{n}(g) = ∑_{j=0}^{∞} \frac{E[g(−b_{n}j + X_{s}^{n∗})𝕀 (σ_{1}^{n∗} > s) ]_{s=t−σ_{j}^{n}}}{E[𝕀(σ_{1}^{n∗} > s)]_{s=t−σ_{j}^{n}}} 𝕀 (σ_{j}^{n} ≤ t < σ_{j+1}^{n}) ,  (21)

where {σ_{j}^{n}, j ≥ 0} is defined by

σ_{j}^{n} = inf {t > 0 s. t. X_{t}^{n} ≤ −b_{n}j}

and represents a renewal process that coincides with the local time L_{t}^{n} up to a space scaling, and X^{n∗} is a process with the same law as X^{n} and σ_{1}^{n∗} is its first exit time from the set (−b_{n}, ∞). Note that, when t ∈ [σ_{j}^{n}, σ_{j+1}^{n}), the stopping time σ_{j}^{n} can be written as

σ_{j}^{n} = sup{u ≤ t s. t. L_{u}^{n} < L_{t}^{n}},

and therefore (21) can be rewritten as

π_{t}^{n}(g) = \frac{E[g(−l + X_{s}^{n})𝕀 (σ_{1}^{n} > s) ]}{E[𝕀(σ_{1}^{n} > s)]}|_{l=L_{t}^{n}, s=ξ_{t}^{n}}

where, ξ_{t}^{n} is defined in (10), i.e.

ξ_{t}^{n} = γ_{t}(L^{n}) = t − sup{u ≤ t s. t. L_{u}^{n} < L_{t}^{n}}.

Equivalently we can rewrite

π_{t}^{n}(g) = E[g(−l + X_{s}^{n})/{L_{s}^{n} < b_{n}}]|_{l=L_{t}^{n}, s=ξ_{t}^{n}}.  (22)

Then, denoting by Σ^{n}(s, l) the probability measure such that

Σ^{n}(s, l; g) = \frac{E[g(−l + X_{s}^{n})𝕀 (σ_{1}^{n} > s) ]}{E[𝕀(σ_{1}^{n} > s)]}  (23)

(21) can be shortly written as

π_{t}^{n}(g) = E[g(X_{t}^{n})/G_{t}^{n}] = Σ^{n}(ξ_{t}^{n}, L_{t}^{n}; g).  (24)

Remark 3.1. Note that Σ^{n}(s, l) depends also on the probability measure P^{n}, i.e. Σ^{n}(s, l) = Σ_{P^{n}}^{n}(s, l). This dependence will be emphasized when necessary.

By taking into account that

π̂_{t}^{n}(g) = E[g(X_{t}^{n} + L_{t}^{n})/G_{t}^{n}] = E[g(X_{t}^{n} + m)/G_{t}^{n}]|_{m=L_{t}^{n}}

the filter can be written as

π̂_{t}^{n}(g) = E[g(X_{t}^{n} + L_{t}^{n})/G_{t}^{n}] = Σ̂^{n}(ξ_{t}^{n}; g),  (25)

where

Σ̂^{n}(s; g) = Σ^{n}(s, 0; g) = \frac{E[g(X_{s}^{n})𝕀 (σ_{1}^{n} > s) ]}{E[𝕀(σ_{1}^{n} > s)]}.  (26)

We end this Section by noting that, when the process X̃^{n} = Ṽ n ; Z̃_{t}^{n} satisfies the assumptions H1, H2 and H3 stated in Section 2, then Proposition 2.3 easily provides the explicit expressions of Σ^{n}(s, l) and Σ̂^{n}(s)

Σ^{n}(s, l; g) = \frac{∑_{k=1}^{∞} E[𝕀 (M_{1}^{n} ≥ k) g(b_{n}Ṽ_{k−1}^{n} − l)](F_{k−1}^{n}(s) − F_{k}^{n}(s))}{∑_{m=1}^{∞} P (M_{1}^{n} = m)(1 − F_{m}^{n}(s))}  (27)

Σ̂^{n}(s; g) = \frac{∑_{k=1}^{∞} E[𝕀 (M_{1}^{n} ≥ k) g(b_{n}Ṽ_{k−1}^{n})](F_{k−1}^{n}(s) − F_{k}^{n}(s))}{∑_{m=1}^{∞} P (M_{1}^{n} = m)(1 − F_{m}^{n}(s))},  (28)

where M_{1}^{n}, F_{k}^{n} have a similar meaning as M_{1}, F_{k} in (19). In particular it is interesting to note that if F̃_{1}^{n} denotes the distribution function of

σ̃_{1}^{n} = inf {t > 0 s. t. X̃_{t}^{n} ≤ −1} ,

then σ_{1}^{n} = σ̃_{1}^{n}/a_{n}, and therefore F_{1}^{n}(s) = F̃_{1}^{n}(a_{n}s), and F_{k}^{n}(s) = F̃_{k}^{n}(a_{n}s), where F̃_{k}^{n} is the k−fold convolution of F̃_{1}^{n}.

The next three sections are devoted to the problem of finding the limit of the filter (24) for the random walk (or of the filter (25) for the reflected random walk), or some good approximation for it, in the case when the rescaled sequence X_{t}^{n} converges to a Brownian motion. For the above models we investigate whether the limit of the filter is the corresponding filter of the limits.

4 The interpolating Brownian motion model

In this Section we study the case of a continuous time random walk arising when a Brownian motion W is approximated with a sequence of processes W^{n}. The processes W^{n} are defined on the probability space of W, and are obtained pathwise by an interpolation procedure. For this model we are able to get a strong convergence result for the filter. We start by introducing the approximating model, then we show the convergence result. Successively we discuss how the approximating models W^{n} fall into the frame of the previous sections. In particular, when the process W is a standard Brownian motion the processes W^{n} corresponds to the case examined at the end of the previous section, with scaling parameters a_{n} = 2^{2n} and b_{n} = 1/2^{n}, p_{n} = q_{n} = 1/2, F̃_{1}^{n} = F̃_{1}, where

F̃_{1}(t) = 4 ∑_{j=0}^{+∞}(−1)^{j} \frac{1}{\sqrt{2π}} ∫ +∞_{\frac{(2j+1)}{\sqrt{t}}} exp (−\frac{x^{2}}{2} ) dx.  (29)

The basic idea is to approximate the state W by the stepwise interpolation of the random points where W hits a uniform grid and consider as approximating observation the local time of the approximating state. This procedure is a deterministic one and therefore we describe it in the deterministic case.

Let z ∈ D_{ℝ}[0, +∞) and let h ∈ ℝ^{+} be a fixed threshold. Consider the sequence {τ_{k}^{h}(z), k ≥ 0}

\begin{cases}τ_{0}^{h}(z) = 0 \\ τ_{k}^{h}(z) = inf \{t > τ_{k−1}^{h}(z) : |z(t) − z(τ_{k−1}^{h}(z))| > h\} , k ≥ 1, \end{cases}  (30)

and the function z^{h} ∈ D_{ℝ}[0, +∞)

z^{h}(t) = ∑_{k=0}^{∞} 𝕀_{[τ_{k}^{h}(z),τ_{k+1}^{h}(z))}(t)z(τ_{k}^{h}(z)).  (31)

We need the following result.

Lemma 4.1. Let z ∈ D_{ℝ}[0, +∞). Then, for each t ∈ ℝ^{+}

|z^{h}(t) − z(t)| ≤ h,  (32)

and

z^{h} →_{h→0} z and ℓ(z^{h}) →_{h→0} ℓ(z)

w.r.t. the topology of the uniform convergence, where the functional ℓ is defined by (1).

Proof. Observing that |z^{h}(t) − z(t)| = |z(τ_{k}^{h}(z)) − z(t)|, for t ∈ [τ_{k}^{h}(z), τ_{k+1}^{h}(z)), and that τ_{k}^{h}(z) ↑ ∞, the bound (32) and the uniform convergence of z^{h} to z follow. The continuity of the functional ℓ implies the other limit.

Remark 4.2. When z is a continuous function, the process ℓ(z^{h}) admits the representation

ℓ_{t}(z^{h}) = 0 ∨ (−z_{0}) − ∑_{j=0}^{∞} z(σ_{j}^{h}(z))𝕀_{[σ_{j}^{h}(z),σ_{j+1}^{h}(z))}(t),

where

σ_{j}^{h}(z) = inf {t s.t. z(t) − z_{0} ≤ −jh} = inf {τ_{k}^{h}(z) s.t. z(τ_{k}^{h}(z)) − z_{0} ≤ −jh}.  (33)

When furthermore z_{0} = 0, then z(σ_{j}^{h}(z)) = −jh and

ℓ_{t}(z^{h}) = ∑_{j=0}^{∞} jh𝕀(σ_{j}^{h}(z) ≤ t < σ_{j+1}^{h}(z)) = ∑_{j=0}^{∞} h𝕀(σ_{j}^{h}(z) ≤ t).  (34)

We now apply this approximating procedure to the Brownian motion W_{t}.

The local time Λ_{t} of the process W_{t} is Λ_{t} = ℓ_{t}(W), where ℓ is the deterministic functional defined by (1). Observe that W_{0} = 0 implies

Λ_{t} = − inf_{0≤s≤t} (W_{s}).  (35)

Fix now the sequence of thresholds h_{n} = \frac{1}{2^{n}}, and consider the stopping times τ_{k}^{n} := τ_{k}^{h}(W), when using h = h_{n} = \frac{1}{2^{n}} in (30). Then the approximating signal/observation process (W^{n}, Λ^{n}) is a D_{ℝ^{2}}[0, +∞)-valued process, where

W_{t}^{n} = ∑_{k=0}^{∞} W (τ_{k}^{n}) 𝕀_{[τ_{k}^{n},τ_{k+1}^{n})}(t), and Λ_{t}^{n} = ℓ_{t}(W^{n}).  (36)

Note that Lemma 4.1 provides the following convergence result.

Lemma 4.3. Let (W^{n}, Λ^{n}) be defined as in (36). Then, for each t ∈ ℝ^{+}

|W_{t}^{n} − W_{t}| ≤ \frac{1}{2^{n}} ,  (37)

and

(W^{n}, Λ^{n}) = (W^{n}, ℓ(W^{n})) →_{n→∞} (W, Λ) = (W, ℓ(W)) a.s.,

w.r.t. the topology of the uniform convergence.

By (34)

Λ_{t}^{n} = ∑_{j=0}^{∞} \frac{1}{2^{n}} 𝕀(σ_{j}^{n} ≤ t),  (38)

where

σ_{j}^{n} = inf \{t s.t. W_{t} ≤ −\frac{j}{2^{n}} \} = inf \{t s.t. Λ_{t} ≥ \frac{j}{2^{n}} \}.  (39)

Moreover, with the above choice of the threshold, the n-th grid is generated by considering the dyadic intervals of rank n. Then in the passage from the n-th grid to the (n + 1)-th grid each threshold is split into two parts, and therefore σ_{2j}^{n+1} = σ_{j}^{n}. This property is decisive since it guarantees that, for any t, {G_{t}^{n} = F_{t}^{Λ^{n}}, n ∈ ℕ} is an increasing family of σ-algebras, with G_{t}^{n} ↑ F_{t}^{Λ} (see Lemma 2.3 of [9], where the process Λ_{t}^{n} is defined as in (38)). The last fact allows us to show the claimed strong convergence result, which is a slight generalization of Theorem 2.4 of [9].

Theorem 4.4. Let g ∈ C_{b}(ℝ) uniformly continuous. Then

π_{t}^{n}(g) = E[g(W_{t}^{n})/G_{t}^{n}] → π_{t}(g) = E[g(W_{t})/F_{t}^{Λ}], a.s. and in L^{1},  (40)

where E[g(W_{t})/F_{t}^{Λ}] is given in (5).

Proof. Observe that

|E[g(W_{t})/F_{t}^{Λ}] − E[g(W_{t}^{n})/G_{t}^{n}]| ≤ |E[g(W_{t})/F_{t}^{Λ}] − E[g(W_{t})/G_{t}^{n}]|+
+ |E[g(W_{t})/G_{t}^{n}] − E[g(W_{t}^{n})/G_{t}^{n}]|.

As in Theorem 2.4 of [9], we apply Doob’s convergence Theorem to the discrete time martingale E[g(W_{t})/G_{t}^{n}] and get

E[g(W_{t})/G_{t}^{n}] → E[g(W_{t})/F_{t}^{Λ}]  a.s. and in L^{1},  (41)

and therefore the first term converges to zero. As far as the second term is concerned, we have

|E[g(W_{t})/G_{t}^{n}] − E[g(W_{t}^{n})/G_{t}^{n}]| ≤ |E[|g(W_{t}) − g(W_{t}^{n})|/G_{t}^{n}]| ≤ ω_{g} (1/2^{n}) → 0,

where ω_{g} (δ) = sup_{|x−y|≤δ} |g(x) − g(y)| is the modulus of continuity of g.

If we consider the space of probability measures on ℝ endowed with the topology of convergence in distribution, then (40) states that the conditional laws π_{t}^{n} converge a.s. to the conditional law π_{t} defined by (5). Indeed the problem is that apparently the set of probability zero depends on g. Nevertheless in ℝ it is possible to find a countable convergence determining class. For example the piecewise linear functions {f_{s,r}(y), s ≤ r ∈ ℚ}, with f_{s,r}(y) = 0 for y ≤ s, f_{s,r}(y) = 1 for y ≥ r. This can be seen directly or by using Theorem 4.5 in [4]. One can get the analogous convergence results for the corresponding queueing model generated by reflecting W^{n}. In particular the conditional laws defined by E[g(W_{t}^{n} + Λ_{t}^{n})/G_{t}^{n}] converge a.s. to the conditional law π̂_{t} defined by (6).

Now we show that the approximating model falls into the frame of the previous sections. The sequence {(T̃_{k}^{n}, Ũ_{k}^{n}), k ≥ 1} is defined as

T̃_{k}^{n} = (τ_{k}^{n} − τ_{k−1}^{n})/2^{2n},  Ũ_{k}^{n} = 2^{n}(W_{τ_{k}^{n}} − W_{τ_{k−1}^{n}}),

which clearly satisfies condition H0. Then

W_{t}^{n} = \frac{1}{2^{n}} Ṽ n ; Z̃_{2^{2n}t}^{n},

where Z̃_{t}^{n} is the renewal process defined by the sequence of i.i.d. interarrival times {T̃_{k}^{n}, k ∈ ℕ}, and Ṽ_{k}^{n} = 2^{n}W_{τ_{k}^{n}}. Therefore, recalling (24), E[g(W_{t}^{n})/G_{t}^{n}] = Σ^{n}(γ_{t}(W^{n}), ℓ_{t}(W^{n}); g), with Σ^{n}(s, l; g) as in (26), and γ_{t} as in (9). Similar result holds also for E[g(W_{t}^{n} + ℓ_{t}(W^{n}))/G_{t}^{n}].

In addition the strong convergence of Theorem 4.4 implies the weak convergence for the filters of any rescaled model X^{n} sharing the same law as W^{n}. As an example we can take X̃^{n} = W^{0} for all n, and X_{t}^{n} = \frac{1}{2^{n}} W_{2^{2n}t}^{0}.

When the drift coefficient of W is zero the random variables T̃_{k}^{n} have common law (see for example [5] page 342)

F^{n}(t) = P (T̃_{k}^{n} ≤ t) = F̃_{1}(2^{2n}a^{2}t)  (42)

where F̃_{1} is defined in (29), and Ũ_{k}^{n} is symmetric and independent of T̃_{k}^{n}, for each k. Therefore Ṽ_{k}^{n} = 2^{n}W_{τ_{k}^{n}} is a symmetric random walk, independent of the renewal process Z̃_{t}^{n}. Then Conditions H1, H2, H3 hold and one can use (27) to define Σ^{n}(s, l). When the drift coefficient is c ≠ 0 the processes Z̃_{t}^{n} and Ṽ_{k}^{n}, defined as above, are not mutually independent, then one cannot use (27). Nevertheless (27), with F^{n} as in (42) above, could be used to get the approximate expression

π̃_{t}^{n}(g) = \frac{E^{P_{0}}[g(W_{t}^{n}) exp(\frac{c}{a^{2}} W_{t}^{n})/G_{t}^{n}]}{E^{P_{0}}[ exp(\frac{c}{a^{2}} W_{t}^{n})/G_{t}^{n}]} = \frac{Σ^{n} (s, l; g(·) exp(\frac{c}{a^{2}} ·))}{Σ^{n} (s, l; exp(\frac{c}{a^{2}} ·))}|_{s=γ_{t}(W^{n}),l=ℓ_{t}(W^{n})} .

Indeed by Kallianpur Striebel formula and Girsanov Theorem

E[g(W_{t}^{n})/G_{t}^{n}] = \frac{E^{P_{0}}[g(W_{t}^{n}) exp(\frac{c}{a^{2}} W_{t})/G_{t}^{n}]}{E^{P_{0}}[ exp(\frac{c}{a^{2}} W_{t})/G_{t}^{n}]}

where P_{0} is equivalent to P, and under P_{0} the process W has drift coefficient zero. Then

E[g(W_{t}^{n})/G_{t}^{n}] = \frac{E^{P_{0}}[g(W_{t}^{n}) exp(\frac{c}{a^{2}} W_{t}^{n}) exp(\frac{c}{a^{2}} (W_{t} − W_{t}^{n}))/G_{t}^{n}]}{E^{P_{0}}[ exp(\frac{c}{a^{2}} W_{t}^{n}) exp(\frac{c}{a^{2}} (W_{t} − W_{t}^{n}))/G_{t}^{n}]},

Moreover, taking into account that |W_{t} − W_{t}^{n}| ≤ \frac{1}{2^{n}}, one can get that

|E[g(W_{t}^{n})/G_{t}^{n}] − π̃_{t}^{n}(g)| ≤ 4 exp (2\frac{|c|}{a^{2}} \frac{1}{2^{n}} )\frac{|c|}{a^{2}} \frac{1}{2^{n}} ∥g∥_{∞}.  (43)

5 The M/M/1 queueing model

In this Section we consider a random walk with exponential interarrival times, and the M/M/1 queue generated by reflecting the random walk. We use the techniques introduced in Section 2 to derive the filter of the M/M/1 queue (and therefore of the random walk) with respect to the local time of the random walk. Moreover, under a suitable set of conditions, which are related to the heavy traffic conditions, we get also the weak limit of the filter of the rescaled system (Theorem 5.1) and an approximation for the filter (Theorem 5.2). In order to do so, we need the key result (13) , which is proven in Proposition 5.5 of Subsection 5.1 in the case when the arrival intensity λ_{n} and the service potential μ_{n} of the M/M/1 queue are equal and constant in n (the symmetric case), and in Proposition 5.10 of Subsection 5.2 for the sequences of M/M/1 queues when {(λ_{n}, μ_{n}), n ∈ ℕ} satisfy conditions C1, C2, C3 below (the general case). The approach of Subsection 5.2 is quite similar to the approach of Subsection 5.1 and does not require the knowledge of the previous results, but it is technically more complicated. This is the reason why, even if the results in the first Subsection can be obtained from the general case, we choose to discuss them separately.

The sequence of random walks we consider is defined by means of the same rule as in (14), namely for each n ∈ ℕ

X̃_{t}^{n} = Ṽ n ; Z̃_{t}^{n} = ∑_{j=1}^{Z̃_{t}^{n}} Ũ_{j}^{n},

where

A1 Z̃_{t}^{n} is a Poisson process with intensity (λ_{n} + μ_{n})

A2 Ṽ_{j}^{n} is defined by Ṽ_{j}^{n} = Ṽ_{j−1}^{n} +Ũ_{j}^{n}, where {Ũ_{j}^{n}, j ∈ ℕ} is a sequence of i.i.d. random variables with P^{n}(Ũ_{k}^{n} = +1) = \frac{λ_{n}}{λ_{n}+μ_{n}} and P^{n}(Ũ_{k}^{n} = −1) = \frac{μ_{n}}{λ_{n}+μ_{n}}.

A3 {Ũ_{k}^{n}, k ∈ ℕ} and Z̃_{t}^{n} are mutually independent.

In this case the interarrival times T̃_{k}^{n} of the renewal process Z̃_{t}^{n} are exponential random variables with expectation 1/(λ_{n} + μ_{n}). Therefore we are in the situation discussed at the end of Section 3, with p_{n} = λ_{n}/(λ_{n} + μ_{n}), F̃_{1}^{n} the distribution function of an exponential random variable of parameter λ_{n} + μ_{n}, moreover the scaling parameters are a_{n} = n and b_{n} = \sqrt{n}, and then F_{k}^{n} is the distribution function Gamma of parameter (k, n(λ_{n} + μ_{n})).

The conditions C1, C2, and C3 are defined as follows.

C1 λ_{n}, μ_{n} > 0

C2 (λ_{n}, μ_{n}) →_{n→+∞} (λ, λ)

C3 \sqrt{n}(λ_{n} − λ) →_{n→+∞} c(1) \sqrt{n}(μ_{n} − λ) →_{n→+∞} c(2)

Condition C1 avoids to consider pure birth or pure death processes, and C3 clearly implies condition C2 and condition

C3* \sqrt{n}(λ_{n} − μ_{n}) →_{n→+∞} c = c(1) − c(2).

We recall that, when λ_{n} < μ_{n}, the set of conditions C1, C2, C3* are known in literature as the heavy traffic conditions, and in this case c ≤ 0. These conditions guarantee the existence of the diffusive limit of the rescaled system, more precisely, the sequence of processes X_{t}^{n} = X̃_{nt}^{n}/\sqrt{n} converges weakly in D_{ℝ}[0, +∞) to a Brownian motion W_{t}, with diffusion coefficient 2λ and drift coefficient c. The reason why we require the stronger condition C3 will be clarified in Remark 5.8. Here we only point out that the above conditions are equivalent to require also the weak convergence of the processes Z_{t}^{n} = (Z̃_{nt}^{n} − 2λnt)/\sqrt{n}.

The solution of the Skorohod problem for the process X̃_{t}^{n} is the pair (Q̃_{t}^{n}, L̃_{t}^{n}), where

Q̃_{t}^{n} = X̃_{t}^{n} + L̃_{t}^{n}

is a M/M/1 queue and L̃_{t}^{n} is the local time for the process X̃_{t}^{n} (see, for instance, [2]). Then, thanks to the weak convergence of X^{n}, a continuous map argument applies to show that

(X_{t}^{n}, Q_{t}^{n}, L_{t}^{n}) = (\frac{X̃_{nt}^{n}}{\sqrt{n}} , \frac{Q̃_{nt}^{n}}{\sqrt{n}} , \frac{L̃_{nt}^{n}}{\sqrt{n}} ) ⇒ (W_{t}, W_{t} + Λ_{t}, Λ_{t}),  (44)

where Λ_{t} = ℓ_{t}(W) is the local time of W_{t}. Indeed the functional ℓ_{t} is continuous with respect to the topology of uniform convergence on bounded intervals of time, and X_{t}^{n} converges to W_{t} with respect to this topology, since W_{t} has continuous trajectories.

The main results are stated in the following theorems which are proven at the end of this section. We recall (see (11) and (12)) that π_{t}^{n} and π̂_{t}^{n} denote the filters of X_{t}^{n} and of Q_{t}^{n} given the filtration G_{t}^{n}, respectively, where, as in Section 3, G_{t}^{n} denotes the filtration generated by L_{t}^{n}.

Theorem 5.1. Assume conditions C1, C2, C3 and A1, A2, A3, then, for any t ≥ 0, π_{t}^{n} converge weakly to π_{t}, and π̂_{t}^{n} converge weakly to π̂_{t}, as random variables with values in the space of probability measures endowed with the topology of weak convergence. In particular, for any t ≥ 0, and for any bounded continuous function g

π_{t}^{n}(g) = E[g(X_{t}^{n})/G_{t}^{n}] ⇒ π_{t}(g) = E[g(W_{t})/F_{t}^{Λ}]  (45)

and

π̂_{t}^{n}(g) = E[g(Q_{t}^{n})/G_{t}^{n}] ⇒ π̂_{t}(g) = E[g(W_{t} + Λ_{t})/F_{t}^{Λ}].  (46)

As explained in the introduction it is interesting to find a good approximation for the filter π̂_{t}^{n} = Σ̂^{n}(ξ_{t}^{n}) which is, at the same time, simpler to handle, and depends on the actually observed trajectory. In the light of the key result (13), and in particular of Proposition 5.10, a natural candidate is

Π̂_{2λ,c}(ξ_{t}^{n}),  (47)

where Π̂_{2λ,c}(s) is defined in (4).

We can show that (47) approximates the filter E^{P^{n}}[g(Q_{t}^{n})/G_{t}^{n}] in the L_{p} (Ω × [0, T])-norm, for each T > 0, p > 0.

Theorem 5.2. Under the same assumption of Theorem 5.1, for all g bounded and continuous and for each T > 0, p > 0

∫_{0}^{T} E|Σ̂^{n}(ξ_{t}^{n}; g) − Π̂_{2λ,c}(ξ_{t}^{n}; g)|^{p}dt ⟶_{n→∞} 0.

The proof of Theorem 5.1, as well as the proof of Theorem 5.2, is based on the representations (24) and (25) for π_{t}^{n} and π̂_{t}^{n}, respectively, and therefore we need a preliminary result concerning the weak convergence of ξ_{t}^{n} = γ_{t}(L^{n}) to ζ_{t} = γ_{t}^{0}(W + Λ) = γ_{t}(Λ), with γ_{t}^{0} and γ_{t} defined in (8) and (9) respectively. We show a slightly stronger result concerning the weak convergence of γ_{t}^{0}(X^{n} + L^{n}) = γ_{t}^{0}(Q^{n}) to ζ_{t}. This result is used later in Section 6.

Proposition 5.3. Assume conditions C1, C2, C3*. For each t > 0

(γ_{t}^{0}(Q^{n}), γ_{t}(L^{n}), L_{t}^{n}) ⇒ (ζ_{t}, ζ_{t}, Λ_{t}).

Proof. Define

η_{t}^{n} = sup{s < t : L_{s}^{n} < L_{t}^{n}}, η_{t} = sup{s < t : Λ_{s} < Λ_{t}},
β_{t}^{n} = sup{s < t : Q_{s}^{n} = 0}, β_{t} = sup{s < t : W_{s} + Λ_{s} = 0},

with η_{t}^{n} = t, η_{t} = t, β_{t}^{n} = t and β_{t} = t if the corresponding set is empty. Note that

η_{t}^{n} = sup{s < t : X_{t}^{n} − X_{s}^{n} < Q_{t}^{n} − Q_{s}^{n}},
η_{t} = sup{s < t : W_{t} − W_{s} < W_{t} + Λ_{t} − W_{s} − Λ_{s}},

Applying the Skorohod representation theorem, we can assume that all the processes live on the same probability space (Ω̄, F̄, P̄), and that

sup_{s≤t}(|X_{s}^{n} − W_{s}| + |Q_{s}^{n} − W_{s} − Λ_{s}|) → 0 P̄-a.s.  (48)

This implies that L^{n} → Λ uniformly in [0, t], P̄-a.s., and

lim inf_{n→∞} η_{t}^{n} ≥ η_{t}.

Then, since

γ_{t}^{0}(Q^{n}) = t − β_{t}^{n}, γ_{t}(L^{n}) = t − η_{t}^{n},

the result is achieved once we prove that the sequence (β_{t}^{n}, η_{t}^{n}) converges P̄-a.s. to (η_{t}, η_{t}) and ζ_{t} = t − η_{t}. Let β_{t}^{∞} = lim sup_{n→∞} β_{t}^{n} and note that Q^{n}(β_{t}^{n}) assumes only the values 0 or \frac{1}{\sqrt{n}}. Then, by (48), W_{β_{t}^{∞}} + Λ_{β_{t}^{∞}} = 0. It follows that

lim sup_{n→∞} β_{t}^{n} ≤ β_{t}.

Moreover, if η_{t}^{n} < t, then Q^{n}(η_{t}^{n}) = 0, if η_{t}^{n} = t, then β_{t}^{n} = t, and it follows that

η_{t}^{n} ≤ β_{t}^{n} for all t, P̄-a.s.

and then

η_{t} ≤ lim inf_{n→∞} η_{t}^{n} ≤ lim sup_{n→∞} β_{t}^{n} ≤ β_{t}, for all t, P̄-a.s.

The proof is achieved since

P̄(η_{t} = β_{t}) = 1

and then ζ_{t} = γ_{t}^{0}(W + Λ) = t − β_{t} = t − η_{t} = t − γ_{t}(Λ). For sake of completeness we prove the previous statement. Note that W_{s} + Λ_{s} = 0 if and only if W_{s} = inf_{r<s} W_{r} ≤ 0. If η_{t} < β_{t}, then for η_{t} ≤ s ≤ β_{t},

W_{t} − W_{s} = W_{t} + Λ_{t} − W_{s} − Λ_{s} ≤ W_{t} + Λ_{t},

and for any rational r satisfying η_{t} < r < β_{t}, we must have

inf_{s≤r} W_{s} = inf_{r≤s≤t} W_{s}.

But P̄{inf_{s≤r} W_{s} = inf_{r≤s≤t} W_{s}} = 0, and hence

P̄{η_{t} < β_{t}} ≤ ∑_{r∈ℚ} P̄{inf_{s≤r} W_{s} = inf_{r≤s≤t} W_{s}} = 0.

Remark 5.4. In the Skorohod space (Ω̄, F̄, P̄) used in the proof of Proposition 5.3, choose a jointly measurable version of ξ_{t}^{n} = γ_{t}(L^{n}) = t − η_{t}^{n}. Then M = {(ω, t) ∈ Ω × [0, T] s.t. ξ_{t}^{n}(ω) ↛ ζ_{t}(ω)} is a zero dP̄ × dt-measure set. Moreover a similar result holds for γ_{t}^{0}(Q^{n}) = t − β_{t}^{n}, namely M_{0} = {(ω, t) ∈ Ω × [0, T] s.t. γ_{t}^{0}(Q^{n})(ω) ↛ ζ_{t}(ω)} is a zero dP̄ × dt-measure set.

Lemma 1.2 combined with (44), guarantees the sequences π_{t}^{n} and π̂_{t}^{n} to be tight. In order to obtain their weak limits, it is crucial to represent E[g(X_{t}^{n})/G_{t}^{n}] and E[g(Q_{t}^{n})/G_{t}^{n}] as in (24) and (25), where they are written as the deterministic functionals Σ^{n}(s, l : g) and Σ̂^{n}(s; g) evaluated at s = ξ_{t}^{n} and l = L_{t}^{n}. Similarly, also the filters π_{t}(g) = E[g(W_{t})/F_{t}^{Λ}] and π̂_{t}(g) = E[g(W_{t} + Λ_{t})/F_{t}^{Λ}] for the limit model are given as the compositions of the deterministic functionals Π_{2λ,c}(s, l; g) and Π̂_{2λ,c}(s; g) evaluated at (s, l) = (ζ_{t}, Λ_{t}) and s = ζ_{t}, respectively (see Theorem 1.1), and moreover Proposition 5.3 proves that (ξ_{t}^{n}, L_{t}^{n}) ⇒ (ζ_{t}, Λ_{t}). These observations induce to investigate whether the sequences of probability measures Σ̂^{n}(s) and Σ^{n}(s, l) (see (26) and (23)) converge to Π̂_{2λ,c}(s) and Π_{2λ,c}(s, l) respectively. In Proposition 5.5 and Proposition 5.10 we show the convergence of Σ̂^{n}(s; g) to Π̂_{2λ,c}(s; g) for the symmetric case and for the general case, respectively. The convergence is uniform on bounded intervals contained in (0, ∞), indeed we prove that, for every g in a convergence determining class, and whenever s_{n} converges to s, with s > 0,

lim_{n→∞} Σ̂^{n}(s_{n}; g) = Π̂_{2λ,c}(s; g),  (49)

which is exactly the key result (13). Then the previous limit is valid for any bounded and continuous function g. We now have all the ingredients for the proof of the main results of this section.

Proof of Theorem 5.1 The weak convergence for filters of the reflected random walk follows since we can use the Skorohod representation probability space as in Proposition 5.3, and in this space, for each t > 0, P̄(ξ_{t}^{n} = γ_{t}(L^{n}) → ζ_{t}) = 1, and, on the other hand P̄{ω : ζ_{t}(ω) > 0} = 1, as observed in Remark 5.4. As a consequence, taking into account (49),

P̄(Σ̂^{n}(ξ_{t}^{n}; g) ⟶_{n→∞} Π̂_{2λ,c}(ζ_{t}; g), for every g : ℝ^{+} → ℝ bounded and continuous) = 1,

and the above property is equivalent to show that π̂_{t}^{n} converges weakly to π̂_{t}. The proof of the weak convergence for the filter of the random walk is similar, since the convergence of Σ^{n}(s_{n}, l_{n}; g) to Π_{2λ,c}(s, l; g) whenever (s_{n}, l_{n}) converges to (s, l), with s > 0 is just a slight extension of the previous result (49), that is for any g : ℝ^{+} → ℝ bounded and uniformly continuous,

Σ^{n}(s_{n}, l_{n}; g) ⟶_{n→∞} Π_{2λ,c}(s, l; g).  (50)

Indeed |Σ^{n}(s_{n}, l_{n}; g) − Σ^{n}(s_{n}, l; g)| ≤ ω_{g}(|l_{n} − l|), where ω_{g}(δ) = sup_{|x−y|≤δ} |g(x) − g(y)| is the modulus of continuity of g, and by (49) Σ^{n}(s_{n}, l; g) = Σ̂^{n}(s_{n}; g_{l}) converge to Π(s, l; g) = Π̂(s; g_{l}), where g_{l}(x) = g(−l+x). The set of bounded and uniformly continuous functions is a convergence determining class and then (50) follows for bounded continuous functions g. Then, using again the Skorohod representation space, we get that P̄((ξ_{t}^{n}, L_{t}^{n}) → (ζ_{t}, Λ_{t}), ζ_{t} > 0) = 1, and

P̄(Σ^{n}(ξ_{t}^{n}, L_{t}^{n}; g) ⟶_{n→∞} Π_{2λ,c}(ζ_{t}, Λ_{t}; g), for every g : ℝ^{+} → ℝ bounded and continuous) = 1.

Therefore π_{t}^{n} converge weakly to π_{t}, and Theorem 5.1 is completely achieved.

□

Proof of Theorem 5.2

The limit we are looking for depends only on the distribution of ξ_{t}^{n}, therefore using the Skorohod representation space (Ω̄, F̄, P̄) as in the proof of Proposition 5.3, the thesis is equivalent to

∫_{0}^{T} EP̄|Σ̂^{n}(ξ_{t}^{n}; g) − Π̂_{2λ,c}(ξ_{t}^{n}; g)|^{p}dt ⟶_{n→∞} 0.

As observed in Remark 5.4, we can assume that

ξ_{t}^{n}(ω) → ζ_{t}(ω) dP̄ × dt − a.e.

Proposition 5.10 implies that

Σ̂^{n}(s_{n}; g) ⟶_{n→∞} Π̂_{2λ,c}(s; g),  (51)

whenever s_{n} → s, s > 0, moreover it is clear that

Π̂_{2λ,c}(s_{n}; g) ⟶_{n→∞} Π̂_{2λ,c}(s; g),

whenever s_{n} → s, s > 0. Combining these results we get

Σ̂^{n}(ξ_{t}^{n}; g) ⟶_{n→∞} Π̂_{2λ,c}(ζ_{t}; g)  and  Π̂_{2λ,c}(ξ_{t}^{n}; g) ⟶_{n→∞} Π̂_{2λ,c}(ζ_{t}; g)  (52)

for each (ω, t) such that ζ_{t}(ω) > 0.

The observation that {(ω, t) ∈ Ω × [0, T] such that ζ_{t}(ω) = 0} is a zero measure set with respect to dP̄ × dt, and an easy application of the dominated convergence theorem imply that

∫_{0}^{T} EP̄ [|Σ̂^{n}(ξ_{t}^{n}; g) − Π̂_{2λ,c}(ζ_{t}; g)|^{p}] → 0, for any p > 0

and

∫_{0}^{T} EP̄ [|Σ̂^{n}(ξ_{t}^{n}; g) − Π̂_{2λ,c}(ξ_{t}^{n}; g)|^{p}] → 0,  for any p > 0.  (53)

□

5.1 The symmetric case

In this subsection we assume that

λ_{n} = μ_{n} = λ.  (54)

Therefore, without loss of generality, we can assume that Z̃_{t}^{n} = Z̃_{t} and Ṽ_{k}^{n} = Ṽ_{k}, where Z̃_{t} is a Poisson process with intensity 2 λ and Ṽ_{k} is a symmetric random walk, defined on a suitable space (Ω, F, P). Then

X̃_{t}^{n} = X̃_{t} = ṼZ̃_{t},

and, with this position, the process X_{t}^{n} = X̃_{nt}/\sqrt{n} depends on the index n just by the timespace scaling, moreover conditions C1, C2 and C3 are satisfied with c(1) = c(2) = 0, and consequently c = 0. In this case the process W_{t} defined by (44) is a driftless Brownian motion with diffusion coefficient 2λ and Λ_{t} is its local time. Without loss of generality we can assume λ = \frac{1}{2}. Otherwise we can use the deterministic change of time t/(2λ) instead of t and consider the sequence of processes X_{t/(2λ)}^{n}, which converges to a standard Brownian motion.

For the sake of notational convenience we denote by Π̂(s) the probability measure defined by

Π̂(s; g) = Π̂_{1,0}(s; g) = Π(s, 0; g) = ∫_{0}^{∞} g(y\sqrt{s})y exp (−\frac{y^{2}}{2} ) dy  (55)

for each g : ℝ^{+} → ℝ bounded and continuous, where Π(s, l) and Π̂_{1,0}(s) are defined in Theorem 1.1.

Proposition 5.5. Let g : ℝ^{+} → ℝ be a bounded continuous function. Under the assumption (54), for each s,

Σ̂^{n}(s; g) ⟶_{n→∞} Π̂(s; g).  (56)

Moreover for any sequence s_{n} converging to s > 0

Σ̂^{n}(s_{n}; g) ⟶_{n→∞} Π̂(s; g).  (57)

In order to prove Proposition 5.5 we need a preliminary result.

Lemma 5.6. For each s > 0 and n ∈ ℕ, let F_{s}^{n} be the distribution function of the random variable X_{s}^{n}. Then

F_{s}^{n}(x) − Φ (\frac{x}{\sqrt{s}}) = o (\frac{1}{\sqrt{ns}}) , uniformly in x ∈ Γ_{n},  (58)

where

Γ_{n} = \{\frac{1}{\sqrt{n}} (z + \frac{1}{2}) , z ∈ ℤ\},

and Φ (x) is the distribution function of a standard normal random variable.

Proof. The random variable X_{s}^{n} can be written as follows

X_{s}^{n} = \frac{1}{\sqrt{n}} ∑_{k=0}^{n} Δ_{s}^{k},

where {Δ_{s}^{k} = X̃_{ks} − X̃_{(k−1)s}, k ∈ ℕ}. Under the assumption stated at the beginning of this Subsection, the sequence {Δ_{s}^{k}, k ∈ ℕ} is a sequence of i.i.d. symmetric random variables, with characteristic function exp{s(cos(u) − 1)}, with first and third moments equal to zero and V ar (Δ_{s}^{1}) = s. Moreover their common distribution function F_{s} is concentrated on the lattice Γ_{n}, as well as the distribution function F_{s}^{n}. Then, since the third moments of Δ_{s}^{k} are zero, by Theorem 2, in [5], Chapter XVI.4 we get

F̂_{s}^{n}(x) − Φ (x) = o (\frac{1}{\sqrt{n}}) , uniformly in x s.t.\frac{x}{\sqrt{s}} ∈ Γ_{n}.  (59)

where F̂_{s}^{n}(x) = F_{s}^{n}(x\sqrt{s}) is the distribution function of \frac{X_{s}^{n}}{\sqrt{s}}. The expansion (59) does not imply immediately the thesis, nevertheless (58) can be proved by repeating and adapting the proof to the case when the limit distribution function is Φ (\frac{x}{\sqrt{s}}) instead of Φ(x).

Remark 5.7. Let s_{n} → s and s > 0, so that there exists n̄ such that s_{n} ≥ \frac{1}{2}s, for any n > n̄. Then, under the assumptions of Lemma 5.6

F_{s_{n}}^{n}(x) − Φ (\frac{x}{\sqrt{s_{n}}} ) = o (\frac{1}{\sqrt{n}}) , uniformly in x ∈ Γ_{n}.  (60)

Proof of Proposition 5.5. We prove in detail only (56). The proof of (57) for the case of a sequence s_{n} converging to s > 0, is treated in a similar way, by using Lemma 5.6 and Remark 5.7.

When s = 0 (56) is trivial, since clearly Π̂(0; g) = Σ̂^{n}(0; g) = g(0). When s > 0, taking into account (22), we note that Σ̂^{n}(s) is also the conditional law of X_{s}^{n} given the event \{L_{s}^{n} < \frac{1}{\sqrt{n}}\}. Then

Σ̂^{n}(s; g) = \frac{Θ^{n}(s; g)}{Θ^{n}(s; 1)},

where 1 denotes the constant function 1(x) = 1, and where Θ^{n}(s) is the measure defined by

Θ^{n}(s; g) = E [g(X_{s}^{n})𝕀\{L_{s}^{n} < \frac{1}{\sqrt{n}}\}] .

We can restrict to the functions g ∈ C^{1}(ℝ) with g′(0) = 0 since this class is convergence determining. Without loss of generality we can also assume that g(0) = 0. In fact

Σ̂^{n}(s; g) = Σ̂^{n}(s; g − g(0)) + g(0).

By an explicit computation we get

Θ^{n}(s; g) = ∑_{k=0}^{∞} g (\frac{k}{\sqrt{n}}) P (X_{s}^{n} = \frac{k}{\sqrt{n}}, min_{0≤u≤s} X_{s}^{n} > \frac{−1}{\sqrt{n}}) .

Moreover, the reflection principle yields

P (X_{s}^{n} = \frac{k}{\sqrt{n}}, min_{0≤u≤s} X_{s}^{n} > \frac{−1}{\sqrt{n}}) =P (X_{s}^{n} = \frac{k}{\sqrt{n}}) − P (X_{s}^{n} = −\frac{k + 2}{\sqrt{n}}) =
=P (X_{s}^{n} = \frac{k}{\sqrt{n}}) − P (X_{s}^{n} = \frac{k + 2}{\sqrt{n}}) ,

and so

Θ^{n}(s; g) = ∑_{k=0}^{∞} g (\frac{k}{\sqrt{n}}) [P (X_{s}^{n} = \frac{k}{\sqrt{n}}) − P (X_{s}^{n} = \frac{k + 2}{\sqrt{n}})] =

=E [g(X_{s}^{n})𝕀\{X_{s}^{n} ≥ 0\} − g(X_{s}^{n} − \frac{2}{\sqrt{n}})𝕀\{X_{s}^{n} − \frac{2}{\sqrt{n}} ≥ 0\}] .  (61)

Set g̃ : ℝ → ℝ, x → g̃(x) = g(x)𝕀(x ≥ 0). Then

Θ^{n}(s; g) = E [g̃(X_{s}^{n}) − g̃(X_{s}^{n} − \frac{2}{\sqrt{n}})] = \frac{2}{\sqrt{n}}E [g̃′(X_{s}^{n} − θ \frac{2}{\sqrt{n}})] ,

where θ is a (0, 1)-valued random variable. Then the sequence of measures Σ̂^{n}(s) admits the following representation

Σ̂^{n}(s; g) = E [g̃′(X_{s}^{n} − θ \frac{2}{\sqrt{n}})] \frac{2}{\sqrt{n}} \frac{1}{Θ^{n}(1)(s)},  (62)

for each g ∈ C^{1}(ℝ) with g′(0) = 0. On the other hand

lim_{n→∞} E [g̃′(X_{s}^{n} − θ \frac{2}{\sqrt{n}})] = E [g̃′(W_{s})] ,  (63)

and an explicit computation yields

E [g̃′(W_{s})] = \frac{1}{\sqrt{2πs}} ∫_{0}^{∞} g′(y) exp (−\frac{y^{2}}{2s}) dy = \frac{1}{\sqrt{2πs}} (−g(0) + ∫_{0}^{∞} g(y)\frac{y}{s} exp (−\frac{y^{2}}{2s}) dy) .

By the change of variable x = \frac{y}{\sqrt{s}} in the integral, and recalling that g(0) = 0 we get

E [g̃′(W_{s})] = \frac{1}{\sqrt{2πs}} (∫_{0}^{∞} g(\sqrt{s}x)x exp (−\frac{x^{2}}{2} ) dx) = \frac{1}{\sqrt{2πs}} Π̂(s; g).

Then, by (62), the proof of (56) is completely achieved by showing that

lim_{n→∞} \frac{\sqrt{n}Θ^{n}(s; 1)}{2} = \frac{1}{\sqrt{2πs}} .  (64)

By (61) we get

Θ^{n}(s; 1) = P (X_{s}^{n} ∈ [0, \frac{1}{\sqrt{n}}]) = F_{s}^{n} (\frac{1}{\sqrt{n}}) − F_{s}^{n} (−\frac{1}{\sqrt{n}}) ,  (65)

and, by using the expedient to write the r.h.s. of the above formula as

F_{s}^{n} (\frac{3}{2\sqrt{n}}) − F_{s}^{n} (−\frac{1}{2\sqrt{n}}) ,

we can use the expansion (58), so that

Θ^{n}(s; 1) =Φ (\frac{3}{2} \frac{1}{\sqrt{ns}}) − Φ (−\frac{1}{2} \frac{1}{\sqrt{ns}}) + o (\frac{1}{\sqrt{ns}}) ≃ \frac{2}{\sqrt{ns}}Φ′(γ_{n}^{s}) + o (\frac{1}{\sqrt{ns}}) ,

where γ_{n}^{s} ∈ (−\frac{1}{2} \frac{1}{\sqrt{ns}}, \frac{3}{2} \frac{1}{\sqrt{ns}}). Then

lim_{n→∞} \frac{\sqrt{n}Θ^{n}(s; 1)}{2} = lim_{n→∞} \frac{1}{\sqrt{s}}Φ′(γ_{n}^{s}) + o (1) = \frac{1}{\sqrt{2πs}}.

□

5.2 The general case

In this subsection our aim is to prove Theorem 5.1 in the general case defined by conditions C1, C2, C3 and A1, A2, A3, with the same space-time scaling. Observe that the processes X̃_{t}^{n} and Z̃_{t}^{n} can be represented as

X̃_{t}^{n} = Ã_{t}^{n} − Ñ_{t}^{n}, Z̃_{t}^{n} = Ã_{t}^{n} + Ñ_{t}^{n}

where, if τ̃_{k}^{n} are the jump times of Z̃^{n},

Ã_{t}^{n} = ∑_{k=0}^{∞} 𝕀(Ũ_{k}^{n} = 1)𝕀(τ̃_{k}^{n} ≤ t)  (66)

Ñ_{t}^{n} = ∑_{k=0}^{∞} 𝕀(Ũ_{k}^{n} = −1)𝕀(τ̃_{k}^{n} ≤ t).  (67)

The processes Ã^{n} and Ñ^{n} are mutually independent Poisson processes with intensities λ_{n} and μ_{n} respectively. Indeed Ã^{n} and Ñ^{n} do not have common jumps and they are Poisson processes with parameters λ_{n}, μ_{n} respectively, and the independence follows by Watanabe’s Theorem (see, for instance, [2]). Then, without loss of generality, we can assume that all the above processes are defined on the same measurable space, but with different probability measures P^{n}. We can use the measurable space (Ω, F) used in Subsection 5.1 for the symmetric case, and take again X̃_{t}^{n} = X̃_{t} = ṼZ̃_{t}, where, under the measure P, the process Z̃_{t} is a Poisson process Z̃_{t} of intensity 2λ and Ṽ_{k} = ∑_{j=1}^{k} Ũ_{j} is a symmetric random walk. Starting from the processes X̃_{t} and Z̃_{t}, and in analogy with (66) and (67), we can define the process Ã_{t} as the process counting the positive jumps of X̃_{t}, and the process Ñ_{t} as the process counting the negative jumps of X̃_{t}.

On (Ω, F) we consider the filtration {F_{t}^{n}, t ∈ [0, T]} generated by the time-rescaled processes (Â_{t}^{n}, N̂_{t}^{n}) = (Ã_{nt}, Ñ_{nt}), and the probability measure P^{n}, absolutely continuous with respect to P, such that

\frac{dP^{n}}{dP} |_{F_{t}^{n}} = L_{t}^{n} = (\frac{λ_{n}}{λ} )Â_{t}^{n} exp \{−n (λ_{n} − λ) t\} (\frac{λ_{n}}{λ} )N̂_{t}^{n} exp \{−n (μ_{n} − μ) t\}  (68)

Under the measure P, the processes Â_{t}^{n}, N̂_{t}^{n} are mutually independent Poisson processes with intensities nλ, nλ while (see [2], Chapter VIII) under P^{n} the processes Â_{t}^{n}, N̂_{t}^{n} are mutually independent Poisson processes with intensities nλ_{n}, nμ_{n}. Moreover, in the probability space (Ω, F, P^{n}), the conditions A1, A2, A3 are satisfied with Z̃_{t}^{n} = Z̃_{t} = Ã_{t} + Ñ_{t}, Ũ_{j}^{n} = Ũ_{j}.

From now to the end of this Section, we denote by E^{P} and E^{P^{n}} the expectations with respect to the probability measures P and P^{n}, respectively.

By Kallianpur Striebel formula, we get that

E^{P^{n}}[g(X_{t}^{n})/G_{t}^{n}] = \frac{E^{P} [g(X_{t}^{n})L_{t}^{n}/G_{t}^{n}]}{E^{P} [L_{t}^{n}/G_{t}^{n}]}.  (69)

Moreover, setting

\begin{cases}Ẑ_{t}^{n} = Â_{t}^{n} + N̂_{t}^{n} \\ X̂_{t}^{n} = Â_{t}^{n} − N̂_{t}^{n}, \end{cases}  (70)

the rescaled processes are X_{t}^{n} = \frac{X̂_{t}^{n}}{\sqrt{n}} and Z_{t}^{n} = \frac{Ẑ_{t}^{n}−2nλt}{\sqrt{n}}, and under the measure P the sequence of processes (X_{t}^{n}, Z_{t}^{n}) converge weakly in D_{ℝ^{2}}([0, ∞)) to two independent Brownian motions (W_{t}, B_{t}) := (W_{t}^{A} − W_{t}^{N}, W_{t}^{A} + W_{t}^{N}), indeed

X_{t}^{n} = \frac{X̂_{t}^{n}}{\sqrt{n}} = \frac{Ã_{nt} − Ñ_{nt}}{\sqrt{n}} = \frac{Ã_{nt} − nλt}{\sqrt{n}} − \frac{Ñ_{nt} − nλt}{\sqrt{n}} ⇒ W_{t}^{A} − W_{t}^{N},

Z_{t}^{n} = \frac{Ẑ_{t}^{n} − 2nλt}{\sqrt{n}} = \frac{Ã_{nt} − nλt}{\sqrt{n}} + \frac{Ñ_{nt} − nλt}{\sqrt{n}} ⇒ W_{t}^{A} + W_{t}^{N},

where W_{t}^{A} and W_{t}^{N} are clearly independent Brownian motion. The last property implies the independence of W and B. Therefore it is natural to get an alternative expression of L_{t}^{n} in terms of the processes X_{t}^{n} and Z_{t}^{n}. Taking into account that, by (70)

\begin{cases}Â_{t}^{n} = (Ẑ_{t}^{n} + X̂_{t}^{n})/2 \\ N̂_{t}^{n} = (Ẑ_{t}^{n} − X̂_{t}^{n})/2 \end{cases}

and that

log(L_{t}^{n}) = log (\frac{λ_{n}}{λ} ) Â_{t}^{n} − n (λ_{n} − λ) t + log (\frac{μ_{n}}{λ} ) N̂_{t}^{n} − n (μ_{n} − λ) t,

we get immediately that ^{2}

log(L_{t}^{n}) = c_{n}X_{t}^{n} + d_{n}Z_{t}^{n} + e_{n}t,  (71)

where

c_{n} = \frac{\sqrt{n}}{2} [log (\frac{λ_{n}}{λ} ) − log (\frac{μ_{n}}{λ} )]  (72)
d_{n} = \frac{\sqrt{n}}{2} [log (\frac{λ_{n}}{λ} ) + log (\frac{μ_{n}}{λ} )]  (73)
e_{n} = n [log (\frac{λ_{n}}{λ} ) + log (\frac{μ_{n}}{λ} )] λ − n (λ_{n} + μ_{n} − 2λ) .  (74)

Therefore (69) can be rewritten as

E^{P^{n}}[g(X_{t}^{n})/G_{t}^{n}] = \frac{E^{P} [g(X_{t}^{n}) exp(c_{n}X_{t}^{n}) exp(d_{n}Z_{t}^{n})/G_{t}^{n}]}{E^{P} [ exp(c_{n}X_{t}^{n}) exp(d_{n}Z_{t}^{n})/G_{t}^{n}]}.  (75)
log(L_{t}^{n}) = log ( \frac{λ_{n}}{λ} ) (Ẑ_{t}^{n} + X̂_{t}^{n})/2 − n (λ_{n} − λ) t + log ( \frac{μ_{n}}{λ} ) (Ẑ_{t}^{n} − X̂_{t}^{n})/2 − n (μ_{n} − λ) t,

= \frac{1}{2} [log ( \frac{λ_{n}}{λ} ) + log ( \frac{μ_{n}}{λ} )] Ẑ_{t}^{n} + \frac{1}{2} [log ( \frac{λ_{n}}{λ} ) − log ( \frac{μ_{n}}{λ} )] X̂_{t}^{n}

= \frac{\sqrt{n}}{2} [log ( \frac{λ_{n}}{λ} ) + log ( \frac{μ_{n}}{λ} )] \frac{Ẑ_{t}^{n} − 2nλt}{\sqrt{n}} + \frac{\sqrt{n}}{2} [log ( \frac{λ_{n}}{λ} ) − log ( \frac{μ_{n}}{λ} )] \frac{X̂_{t}^{n}}{\sqrt{n}}

+ [log ( \frac{λ_{n}}{λ} ) + log ( \frac{μ_{n}}{λ} )] nλt − n (λ_{n} + μ_{n} − 2λ) t

2Indeed

− n (λ_{n} + μ_{n} − 2λ) t

Under conditions C1, C2 and C3, the sequence (c_{n}, d_{n}) converges to (c̄, d̄), where c̄ = c/(2λ) (see Lemma 5.12 at the end of the section). If we substitute in the right hand side of the above expression the formal limits we get

\frac{E[g(W_{t}) exp(c̄W_{t}) exp(d̄B_{t})/F_{t}^{Λ}]}{E[ exp(c̄W_{t}) exp(d̄B_{t})/F_{t}^{Λ}]} = \frac{E[g(W_{t}) exp(c̄W_{t})/F_{t}^{Λ}]}{E[ exp(c̄W_{t})/F_{t}^{Λ}]} = Π̂_{2λ,c}(ζ_{t}; g),  (76)

where the first equality holds since F_{t}^{Λ} ⊂ F_{t}^{W} and the processes W and B are independent, while the second equality follows by the fact that W_{t} has drift zero, variance 2λ, the value of the limit c̄ and by using Kallianpur-Striebel formula again.

Remark 5.8. We also observe that under condition C3 the sequence e_{n} converges to ē = −(c_{1}^{2} + c_{2}^{2})/2λ (see Lemma 5.12 again). Though this fact is irrelevant for our purposes, it is interesting to note that the convergence of (c_{n}, d_{n}, e_{n}) to (c̄, d̄, ē) could be used to prove that the laws under P^{n} of the pair of processes (X_{t}^{n}, Z_{t}^{n}) converge weakly in D_{ℝ^{2}}([0, ∞)) to a pair of independent Brownian motions (W_{t}, B_{t}) with drift c = c(1)−c(2) and d = c(1)+c(2) respectively, and both with variance 2λ. This can be seen also directly by observing that

X_{t}^{n} = \frac{X̂_{t}^{n}}{\sqrt{n}} = \frac{Ã_{nt} − Ñ_{nt}}{\sqrt{n}} = \frac{Ã_{nt} − nλ_{n}t}{\sqrt{n}} − \frac{Ñ_{nt} − nμ_{n}t}{\sqrt{n}} + \sqrt{n}(λ_{n} − μ_{n})t,

Z_{t}^{n} = \frac{Ẑ_{t}^{n} − 2nλt}{\sqrt{n}} = \frac{Ã_{nt} − nλ_{n}t}{\sqrt{n}} + \frac{Ñ_{nt} − nμ_{n}t}{\sqrt{n}} + \sqrt{n}(λ_{n} + μ_{n} − 2λ)t,

that under P^{n}

(\frac{Ã_{nt} − nλ_{n}t}{\sqrt{n}}, \frac{Ñ_{nt} − nμ_{n}t}{\sqrt{n}}) ⇒ (W_{t}^{A,0}, W_{t}^{N,0}),

and to get the weak convergence of X^{n} it is necessary condition C3*, i.e. that \sqrt{n}(λ_{n} − μ_{n}) converges to c, while to get the weak convergence of Z^{n} it is necessary that \sqrt{n}(λ_{n} + μ_{n} − 2λ) converges to a constant d, which is implied by condition C3 with d = c(2) + c(1). Then

W_{t} = W_{t}^{A,0} − W_{t}^{N,0} + ct and B_{t} = W_{t}^{A,0} + W_{t}^{N,0} + dt.

Finally we observe that

\sqrt{n}(λ_{n} − μ_{n}) → c, \sqrt{n}(λ_{n} + μ_{n} − 2λ) → d

is equivalent to

2\sqrt{n}(λ_{n} − λ) → c + d, 2\sqrt{n}(μ_{n} − λ) → d − c,

which is exactly condition C3, with c(1) = (c + d)/2 and c(2) = (d − c)/2.

The above considerations leads to a heuristic proof of our main result, nevertheless we do not formalize the above heuristic reasoning to get the proof. We instead use the following expression for the filter of the queue

E^{P^{n}}[g(Q_{t}^{n})/G_{t}^{n}] = \frac{E^{P} [ g(X_{s}^{n}) exp(c_{n}X_{s}^{n}) exp(d_{n}Z_{s}^{n})𝕀(σ_{1}^{n} > s)]}{E^{P} [ exp(c_{n}X_{s}^{n}) exp(d_{n}Z_{s}^{n})𝕀(σ_{1}^{n} > s)]}|_{s=ξ_{t}^{n}} ,  (77)

which holds since (25), i.e. E^{P^{n}}[g(Q_{t}^{n})/G_{t}^{n}] = Σ̂_{P^{n}}^{n}(ξ_{t}^{n}; g), where, by (26) and the definition of P^{n}

Σ̂_{P^{n}}^{n}(s; g) = \frac{E^{P^{n}}[g(X_{s}^{n})𝕀 (σ_{1}^{n} > s) ]}{E^{P^{n}}[𝕀(σ_{1}^{n} > s)]} = \frac{E^{P} [L_{s}^{n}g(X_{s}^{n})𝕀 (σ_{1}^{n} > s) ]}{E^{P} [L_{s}^{n}𝕀(σ_{1}^{n} > s)]}

and finally by using (71) we get (77).

As we did for the symmetric case, in order to get the limit of previous filter the idea is to show the convergence of the function Σ̂_{P^{n}}^{n}(s; g)

Σ̂_{P^{n}}^{n}(s; g) = \frac{E^{P} [ g(X_{s}^{n}) exp(c_{n}X_{s}^{n}) exp(d_{n}Z_{s}^{n})𝕀(σ_{1}^{n} > s)]}{E^{P} [ exp(c_{n}X_{s}^{n}) exp(d_{n}Z_{s}^{n})𝕀(σ_{1}^{n} > s)]}.  (78)

Then, a first essential step consists in evaluating

E^{P} [ f(X_{s}^{n}) exp(d_{n}Z_{s}^{n})𝕀(σ_{1}^{n} > s)]

either for f(x) = g_{n}(x) = g(x) exp(c_{n}x) or for f(x) = exp(c_{n}x), and this can be found in the following lemma.

Lemma 5.9. Let f be a function with continuous derivative f′. Then

E^{P} [ f(X_{s}^{n}) exp(d_{n}Z_{s}^{n})𝕀(σ_{1}^{n} > s)] =E^{P} [\frac{2}{\sqrt{n}} 𝕀(X_{s}^{n} ≥ 2/\sqrt{n})f′(X_{s}^{n} − 2θ_{f}^{n}/\sqrt{n}) exp(d_{n}Z_{s}^{n})]
+ E^{P} [𝕀(0 ≤ X_{s}^{n} < 2/\sqrt{n})f(X_{s}^{n}) exp(d_{n}Z_{s}^{n})],

where θ_{f}^{n} is a random variable with values in (0, 1).

Proof. Observe that

E^{P} [ f(X_{s}^{n}) exp(d_{n}Z_{s}^{n})𝕀(σ_{1}^{n} > s)] = E^{P} [ f(X_{s}^{n})𝕀(X_{s}^{n} ≥ 0) exp(d_{n}Z_{s}^{n})𝕀(σ_{1}^{n} > s)]

= E^{P} [ f̃(X_{s}^{n}) exp(d_{n}Z_{s}^{n})] − E^{P} [ f̃(X_{s}^{n}) exp(d_{n}Z_{s}^{n})𝕀(σ_{1}^{n} ≤ s)]

= E^{P} [ \{f̃(X_{s}^{n}) − f̃(X_{s}^{n} − \frac{2}{\sqrt{n}})\} exp(d_{n}Z_{s}^{n})],  (79)

where f̃(x) = f(x)𝕀(x ≥ 0), and where we apply the reflection principle in order to get the last equality. Indeed, if X̄_{s}^{n} is the process^{3} obtained by reflecting X_{s}^{n} at time σ_{1}^{n}, then

f(X_{s}^{n})𝕀(X_{s}^{n} ≥ 0) exp(d_{n}Z_{s}^{n})𝕀(σ_{1}^{n} ≤ s) = f(−X̄_{s}^{n} − 2/\sqrt{n})𝕀(X̄_{s}^{n} ≤ −2/\sqrt{n}) exp(d_{n}Z_{s}^{n}) 𝕀(σ̄_{1}^{n} ≤ s)

= f(−X̄_{s}^{n} − 2/\sqrt{n})𝕀(X̄_{s}^{n} ≤ −2/\sqrt{n}) exp(d_{n}Z_{s}^{n}),  (80)

since

(i) σ_{1}^{n} ≤ s if and only if σ̄_{1}^{n} ≤ s, where σ̄_{1}^{n} = inf\{u such that X̄_{u}^{n} ≤ −1/\sqrt{n}\}, (ii) if σ̄_{1}^{n} ≤ s then X_{s}^{n} + 1/\sqrt{n} = −1/\sqrt{n} − X̄_{s}^{n}, and therefore, when σ̄_{1}^{n} ≤ s, (iii) X_{s}^{n} ≥ 0 if and only if −2/\sqrt{n} ≥ X̄_{s}^{n}, which implies σ̄_{1}^{n} ≤ s, and (79) follows by (80), taking into account that (−X̄_{s}^{n}, Z_{s}^{n}) has the same law as (X_{s}^{n}, Z_{s}^{n}) under P.

3If (Â_{s}^{n}, N̂_{s}^{n}) are the independent Poisson processes used in constructing (X_{s}^{n}, Z_{s}^{n}), then the reflected process X̄_{s}^{n} can be obtained as X̄_{s}^{n} = (Ā_{s}^{n} − N̄_{s}^{n})/\sqrt{n} , where (Ā_{s}^{n}, N̄_{s}^{n}) as (Â_{s}^{n}, N̂_{s}^{n}) for s < σ_{1}^{n}, and as (Â_{σ_{1}^{n}}^{n} + (N̂_{s}^{n} − N̂_{σ_{1}^{n}}^{n}), N̂_{σ_{1}^{n}}^{n} + (Â_{s}^{n} − Â_{σ_{1}^{n}}^{n})) for s ≥ σ_{1}^{n}.

Moreover, by using the identity f(x) − f(x − 2/\sqrt{n}) = f′(x − 2v_{f}^{n}(x)/\sqrt{n})\frac{2}{\sqrt{n}}, for v_{f}^{n}(x) ∈ (0, 1), we observe that

f̃(x) − f̃(x − 2/\sqrt{n}) =𝕀(x ≥ 0)f(x) − 𝕀(x ≥ 2/\sqrt{n})f(x − 2/\sqrt{n}) =
=𝕀(x ≥ 2/\sqrt{n})(f(x) − f(x − 2/\sqrt{n})) + 𝕀(0 ≤ x < 2/\sqrt{n})f(x) =
=(2/\sqrt{n}) 𝕀(x ≥ 2/\sqrt{n})f′(x − 2v_{f}^{n}(x)/\sqrt{n}) + 𝕀(0 ≤ x < 2/\sqrt{n})f(x).

If we substitute x with X_{s}^{n}, we define θ_{f}^{n} = v_{f}^{n}(X_{s}^{n}), and finally we multiply by exp(d_{n}Z_{s}^{n}), then we get the result by taking the expectation with respect to P and by using (79).

We are now ready to prove the main result of this subsection.

Proposition 5.10. Let g : ℝ^{+} → ℝ be a bounded continuous function. Under the assumptions A1, A2 and A3 and conditions C1, C2 and C3 for each s,

Σ̂_{P^{n}}^{n}(s; g) ⟶_{n→∞}Π̂_{2λ,c}(s; g).  (81)

Moreover whenever s_{n} → s, with s > 0

Σ̂_{P^{n}}^{n}(s_{n}; g) ⟶_{n→∞} Π̂_{2λ,c}(s; g).  (82)

Proof. The case s = 0 is trivial. When s > 0, in order to show (81) we multiply the numerator and the denominator of (78) by \sqrt{n}. Without loss of generality, we can assume that g has continuous bounded derivative. Then, by Lemma 5.9, we need to evaluate the limit of

\sqrt{n}E^{P} [\frac{2}{\sqrt{n}} 𝕀(X_{s}^{n} ≥ 2/\sqrt{n})g′_{n}(X_{s}^{n} − 2θ_{g}/\sqrt{n}) exp(d_{n}Z_{s}^{n})]

+ \sqrt{n}E^{P} [𝕀(0 ≤ X_{s}^{n} < 2/\sqrt{n})g_{n}(X_{s}^{n}) exp(d_{n}Z_{s}^{n})],

when g_{n}(x) = g(x) exp(c_{n}x), for the numerator, and then the limit of the denominator follows taking g(x) = 1. Recalling that (under P) (X_{t}^{n}, Z_{t}^{n}) converge weakly in D_{ℝ^{2}}([0, ∞)) to two independent Brownian motions (W_{t}, B_{t}), the limit of the first addend is

2E^{P} [𝕀(0 < W_{s} < ∞)f′(W_{s})) exp (d̄B_{s})]
= 2E^{P} [𝕀(0 < W_{s} < ∞)f′(W_{s})] E^{P} [ exp (d̄B_{s})],

with f(x) = g(x) exp(c̄x). Moreover, as far as the second addend is concerned, in Lemma 5.11 below we prove that

\sqrt{n}E^{P} [𝕀(0 ≤ X_{s}^{n} < 2/\sqrt{n})g_{n}(X_{s}^{n}) exp(d_{n}Z_{s}^{n})] → \frac{2}{\sqrt{2π}\sqrt{2λs}}g(0)E^{P} [ exp (d̄B_{s})].

Taking into account that (under P) W has drift zero and diffusion coefficient 2λ, and that c̄ = \frac{c}{2λ}, E^{P} [𝕀(0 < W_{s} < ∞)f′(W_{s})] can be written as

2 ∫_{0}^{∞} f′(x) \frac{1}{\sqrt{2π}\sqrt{2λs}} exp\{−\frac{1}{2} \frac{x^{2}}{2λs}\}dx

= \frac{2}{\sqrt{2π}\sqrt{2λs}} ([f(x) exp\{−\frac{1}{2} \frac{x^{2}}{2λs}\}]_{0}^{∞} + ∫_{0}^{∞} f(x)\frac{x}{2λs} exp\{−\frac{1}{2} \frac{x^{2}}{2λs}\}dx)

= \frac{2}{\sqrt{2π}\sqrt{2λs}} (−f(0) + ∫_{0}^{∞} f(x)\frac{x}{2λs} exp\{−\frac{1}{2} \frac{x^{2}}{2λs}\}dx)

= \frac{2}{\sqrt{2π}\sqrt{2λs}} (−g(0) + ∫_{0}^{∞} g(x) exp (\frac{c}{2λ}x) \frac{x}{2λs} exp\{−\frac{1}{2} \frac{x^{2}}{2λs}\}dx) .

Then, summarizing, we obtain

\sqrt{n}E^{P} [ g_{n}(X_{s}^{n}) exp(d_{n}Z_{s}^{n})𝕀(σ_{1}^{n} > s)] →

E^{P} [ exp (d̄B_{s})] \frac{2}{\sqrt{2π}\sqrt{2λs}} ∫_{0}^{∞} g(x) exp (\frac{c}{2λ}x) \frac{x}{2λs} exp\{−\frac{1}{2} \frac{x^{2}}{2λs}\}dx,

that, together with (78), yields

lim_{n→∞} Σ_{P^{n}}^{n}(s : g) = lim_{n→∞} \frac{\sqrt{n}E^{P} [ g(X_{s}^{n}) exp(c_{n}X_{s}^{n}) exp(d_{n}Z_{s}^{n})𝕀(σ_{1}^{n} > s)]}{\sqrt{n}E^{P} [ exp(c_{n}X_{s}^{n}) exp(d_{n}Z_{s}^{n})𝕀(σ_{1}^{n} > s)]}

= \frac{∫_{0}^{∞} g(x) exp(c̄x)\frac{x}{2λs} exp\{−\frac{1}{2} \frac{x^{2}}{2λs}\}dx}{∫_{0}^{∞} exp(c̄x)\frac{x}{2λs} exp\{−\frac{1}{2} \frac{x^{2}}{2λs}\}dx}

= \frac{Π(2λs, 0; g(·) exp(\frac{c}{2λ}·))}{Π(2λs, 0; exp(\frac{c}{2λ}·))} = Π̂_{2λ,c}(s; g).

The proof of (82) is similar to the previous one and can be achieved by using the weak convergence of (X_{s}^{n}, Z_{s}^{n}) to (W_{s}, B_{s}) in the uniform norm on bounded intervals, and Lemma 5.11. Lemma 5.11. Under the same conditions of Proposition 5.10, let

q_{n}(s) := E^{P} [𝕀(0 ≤ X_{s}^{n} < 2/\sqrt{n}) g(X_{s}^{n}) exp(c_{n}X_{s}^{n}) exp(d_{n}Z_{s}^{n})],

then

lim_{n→∞} \sqrt{n}q_{n}(s_{n}) = \frac{2}{\sqrt{2π}\sqrt{2λs}} g(0) E^{P} [ exp (d̄B_{s})],  (83)

whenever s_{n} → s, with s > 0. Proof. First of all we observe that

q_{n}(s) = ∑_{h=0}^{∞} P (Ã_{ns} = h)P (Ñ_{ns} = h)g(0)e^{c_{n}0}e^{d_{n} \frac{2h−2nλs}{\sqrt{n}}}

+ ∑_{h=0}^{∞} P (Ã_{ns} = h + 1)P (Ñ_{ns} = h)g(1/\sqrt{n})e^{c_{n} \frac{1}{\sqrt{n}}} e^{d_{n} \frac{2h+1−2nλs}{\sqrt{n}}}

In order to evaluate (83) we observe that \sqrt{n} q_{n}(s_{n}) has the same behaviour as g(0) \sqrt{n}q̄_{n}(s_{n}), where

q̄_{n}(s) := ∑_{h=0}^{∞} P (Ã_{ns} = h)P (Ñ_{ns} = h)e^{d_{n} \frac{2h−2nλs}{\sqrt{n}}}

+ ∑_{h=0}^{∞} P (Ã_{ns} = h + 1)P (Ñ_{ns} = h)e^{d_{n} \frac{2h+1−2nλs}{\sqrt{n}}} ,

as can be immediately seen^{4}. Taking into account that Z̃_{ns} = Ã_{ns} + Ñ_{ns} is a Poisson random variable of parameter 2λns, one can see that^{5}

P (Ã_{ns} = h)P (Ñ_{ns} = h) = P (Z̃_{ns} = 2h)\frac{(2h)!}{h! h!} \frac{1}{2^{2h}}

and that

P (Ã_{ns} = h + 1)P (Ñ_{ns} = h) = P (Z̃_{ns} = 2h + 1)\frac{(2h + 1)!}{(h + 1)! h!} \frac{1}{2^{2h+1}} .

Then, setting

r(k) = \frac{k!}{(k − [k/2])! [k/2]!} \frac{1}{2^{k}} ,

we can rewrite

q̄_{n}(s) = ∑_{k=0}^{∞} P (Z̃_{ns} = k)e^{d_{n} \frac{k−2nλs}{\sqrt{n}}} r(k)

= E^{P} [r(Z̃_{ns}) exp(d_{n} \frac{Z̃_{ns} − 2nλs}{\sqrt{n}})] = E^{P} [r(Z̃_{ns}) exp(d_{n}Z_{s}^{n})]

Now by Stirling formula

\frac{r(m)}{\frac{1}{\sqrt{m}}} → \frac{2}{\sqrt{2π}} , as m → ∞

and therefore, we rewrite

\sqrt{n} q̄_{n}(s) = \sqrt{n} P (Z̃_{ns} = 0) + E^{P} [𝕀(Z̃_{ns} > 0) \frac{r(Z̃_{ns})}{\frac{1}{\sqrt{Z̃_{ns}}}} \frac{1}{\sqrt{\frac{Z̃_{ns}}{n}}} exp (d_{n}Z_{s}^{n})] .

P (Ã_{ns} = h)P (Ñ_{ns} = h) = \frac{(λns)^{h}}{h!} e^{−λns} \frac{(λns)^{h}}{h!} e^{−λns}

= \frac{(λns)^{2h}}{h! h!} e^{−2λns} = \frac{(2λns)^{2h}}{(2h)!} e^{−2λns} \frac{(2h)!}{h! h!} \frac{1}{2^{2h}} = P (Z̃_{ns} = 2h) \frac{(2h)!}{h! h!} \frac{1}{2^{2h}}

P (Ã_{ns} = h + 1)P (Ñ_{ns} = h) = \frac{(λns)^{h+1}}{(h + 1)!} e^{−λns} \frac{(λns)^{h}}{h!} e^{−λns}

= \frac{(λns)^{2h+1}}{(h + 1)! h!} e^{−2λns} = \frac{(2λns)^{2h+1}}{(2h)!} e^{−2λns} \frac{(2h + 1)!}{(h + 1)! h!} \frac{1}{2^{2h+1}} = P (Z̃_{ns} = 2h + 1) \frac{(2h)!}{(h + 1)! h!} \frac{1}{2^{2h+1}}

4Indeed

min (g(0), g(1/\sqrt{n}))\sqrt{n}q̄_{n}(s) ≤ \sqrt{n}q_{n}(s) ≤ e^{c_{n} \frac{1}{\sqrt{n}}} max (g(0), g(1/\sqrt{n}))\sqrt{n}q̄_{n}(s).

5By the following simple calculations:

We are interested in the asymptotic behaviour of \sqrt{n} q̄_{n}(s_{n}), and first of all we note that the sequence nP (Z_{s_{n}}^{n} = 0) converges to zero, as can be seen by direct calculations. Then we observe that for each T > 0, by Kolmogorov inequality

P ( sup_{s≤T} |\frac{Z̃_{ns}}{n} − 2λs| ≥ ε) ≤ \frac{V ar(Z̃_{nT})}{n^{2}ε^{2}} = \frac{2nλT}{n^{2}ε^{2}} .

Furthermore Z_{s}^{n} converge weakly to B_{s} in D_{R}([0, ∞)), w.r.t. the topology of uniform convergence on bounded intervals, the limit process having continuous paths. Therefore the pair (Z̃_{ns}/n, Z_{s}^{n}) converges in D_{R}([0, ∞)) × D_{R}([0, ∞)), each component endowed with the topology of uniform convergence, and then

(\frac{Z̃_{ns}}{n} , Z_{s}^{n}) ⇒ (2λs, B_{s}) , in D_{R^{2}}([0, ∞))

with the topology of uniform convergence on bounded intervals.

By Skorohod theorem we can assume w.l.o.g. that the above pair converges P −a.s., and uniformly on bounded intervals. Then,

\frac{r(Z̃_{ns_{n}})}{\frac{1}{\sqrt{Z̃_{nsn}}}} 𝕀(Z̃_{ns_{n}} > 0) \frac{1}{\sqrt{\frac{Z̃_{nsn}}{n}}} exp (d_{n}Z_{s_{n}}^{n}) → \frac{2}{\sqrt{2π}} \frac{1}{\sqrt{2λs}} exp (d̄B_{s})  P − a.s.  (84)

whenever s_{n} → s, with s > 0. The above convergence is equivalent to the uniform convergence on bounded and compact intervals of (0, ∞), and its proof is straightforward. We only observe

that, if we set h_{n}(s) = \sqrt{Z̃_{ns}/n}, then, for any T > 0, h_{n}(s) converge to h(s) = \sqrt{2λs} uniformly in [0, T], and therefore, for any \underline{h} > 0, 1/h_{n}(s) converge to 1/h(s) uniformly in the set \{s, such that h(s) ≥ \underline{h}\}. Moreover the sequence at the r.h.s. of (84) is uniformly integrable since

sup_{n} E^{P} [(\sqrt{n} r(Z̃_{ns_{n}}) exp (d_{n}Z_{s_{n}}^{n}))^{2}] ≤ L < ∞.

Indeed^{6}

sup_{n} E^{P} [𝕀(Z̃_{ns_{n}} > 0) \frac{n}{Z̃_{ns_{n}}} exp (2 d_{n}Z_{s_{n}}^{n})] ≤ L′ < ∞,

sup_{m} \frac{r(m)}{\frac{1}{\sqrt{m}}} ≤ L′′ < ∞.

Then the thesis is achieved since

lim_{n→∞} \sqrt{n} q̄_{n}(s_{n}) = \frac{2}{\sqrt{2π}} \frac{1}{\sqrt{2λs}}E^{P} [ exp (d̄B_{s})].

E^{P} [𝕀(Z̃_{ns_{n}} > 0) \frac{n}{Z̃_{ns_{n}}} exp (2 d_{n}Z_{s_{n}}^{n})] = ∑_{k=1}^{∞} \frac{n}{k} exp (2d_{n} \frac{k − 2λns}{\sqrt{n}} ) \frac{(2λns_{n})^{k}}{k!} e^{−2λns_{n}}

= exp(−2d_{n}/\sqrt{n}) \frac{1}{2λs_{n}} ∑_{k=1}^{∞} \frac{k + 1}{k} exp (2d_{n} \frac{k + 1 − 2λns}{\sqrt{n}} ) \frac{(2λns_{n})^{k+1}}{(k + 1)!} e^{−2λns_{n}}

≤ exp(−2d_{n}/\sqrt{n}) \frac{1}{λs_{n}} E^{P} [exp(2d_{n}Z_{s_{n}}^{n})]

6Note that which is bounded and converges to \frac{1}{λs} E[exp (2d̄B_{s})]

We end this section with the following elementary technical lemma.

Lemma 5.12. If condition C3 holds, then

lim_{n→∞}(c_{n}, d_{n}, e_{n}) = (c̄, d̄, ē),  (85)

where c_{n}, d_{n} and e_{n} are defined in (72),(73) and (74), and where

c̄ = \frac{c}{2λ} = \frac{c(1) − c(2)}{2λ}, d̄= \frac{d}{2λ} = \frac{c(1) + c(2)}{2λ}, ē = −(c_{1}^{2} + c_{2}^{2})/2λ.

Proof. Set α_{n} = λ_{n} − λ and β_{n} = μ_{n} − λ, and ε_{n} = β_{n} − α_{n} = μ_{n} − λ_{n}. Then,

c_{n} = \frac{\sqrt{n}}{2} [log (\frac{λ_{n}}{λ} ) − log (\frac{μ_{n}}{λ} )] = \frac{\sqrt{n}}{2} log (1 − \frac{ε_{n}}{μ_{n}} )

and

d_{n} = \frac{\sqrt{n}}{2} [log (\frac{λ_{n}}{λ} ) + log (\frac{μ_{n}}{λ} )] = \frac{\sqrt{n}}{2} [log (1 + \frac{α_{n}}{λ} ) + log (1 + \frac{β_{n}}{λ} )]

By expanding log(1 + x) = x + o(x) we get

c_{n} = \frac{\sqrt{n}}{2} [−\frac{ε_{n}}{μ_{n}} + o (\frac{ε_{n}}{μ_{n}} )] ,  d_{n} = \frac{\sqrt{n}}{2} [\frac{α_{n} + β_{n}}{λ} + o(α_{n}) + o(β_{n})] ,

and then

c_{n} = \frac{\sqrt{n}}{2} [\frac{1}{\sqrt{n}} \frac{c}{μ_{n}} + o (\frac{1}{\sqrt{n}})]  d_{n} = \frac{\sqrt{n}}{2} [\frac{1}{\sqrt{n}} \frac{d}{λ} + o(\frac{1}{\sqrt{n}})] ,

since C3 implies that α_{n} = \frac{c(1)}{\sqrt{n}} + o (\frac{1}{\sqrt{n}}) , β_{n} = \frac{c(2)}{\sqrt{n}} + o (\frac{1}{\sqrt{n}}) , ε_{n} = −\frac{c}{\sqrt{n}} + o (\frac{1}{\sqrt{n}}) .

The limit (85) for the first two components follows from the convergence of μ_{n} to λ. Using the expansion log(1 + x) = x − x^{2}/2 + o(x^{2}) one can prove that

e_{n} = n [log (\frac{λ_{n}}{λ} ) + log (\frac{μ_{n}}{λ} )] λ − n (λ_{n} + μ_{n} − 2λ)

= n \{[\frac{α_{n}}{λ} − \frac{1}{2} (\frac{α_{n}}{λ} )^{2} + o (α_{n}^{2}) + \frac{β_{n}}{λ} − \frac{1}{2} (\frac{β_{n}}{λ} )^{2} + o (β_{n}^{2})] λ − (α_{n} + β_{n})\}

= −\frac{1}{2λ} (\sqrt{n}(λ_{n} − λ))^{2} − \frac{1}{2λ} (\sqrt{n}(μ_{n} − λ))^{2} + n o (\frac{1}{n}) → −\frac{1}{2} \frac{c_{1}^{2} + c_{2}^{2}}{λ}

Remark 5.13. From the proof it is easy to see that for the convergence of c_{n} to c̄ we only need conditions C2 and C3*, which imply that the sequence \sqrt{n}ϵ_{n}/μ_{n} converges to c/λ. On the other hand, condition C3* does not assure that d_{n} is converging, and d_{n} can be either bounded or unbounded. Indeed one can take for instance μ_{n} = λ + c(2)/\sqrt{n} + (−1)^{n}b/\sqrt{n} and λ_{n} = λ + c(1)/\sqrt{n} + (−1)^{n}b/\sqrt{n}, so that d_{n} = (c(2) + c(1) + 2(−1)^{n}b)/(2λ) + o (1) is a bounded (but not convergent) sequence, or μ_{n} = λ + c(2)/\sqrt{n} + b/n^{γ} and λ_{n} = λ + c(1)/\sqrt{n} + b/n^{γ}, for 0 < γ < 1/2, so that d_{n} is unbounded, though in either cases conditions C2 and C3* are satisfied.

6 The M/M/1 queueing model: observing the idle time process

In this section we are interested in the conditional law of the M/M/1 queue Q̃_{s}^{n}, when the observation process is the idle time process, i.e.

C̃_{t}^{n} = ∫_{0}^{t} 𝕀 (Q̃_{s}^{n} = 0) ds,

the cumulative time the queue has spent in 0, up to t.

Equivalently one can consider as observation process the bivariate point process (Ĩ_{t}^{n}, B̃_{t}^{n}), where Ĩ_{t}^{n} is the process that counts the times when the system starts an idle period and B̃_{t}^{n} is the process that counts the times when the system starts a busy period, that is

Ĩ_{t}^{n} = ∫_{0}^{t} 𝕀(Q̃_{s−}^{n} = 1)dÑ_{s}^{n}  (86)

B̃_{t}^{n} = ∫_{0}^{t} 𝕀(Q̃_{s−}^{n} = 0)dÃ_{s}^{n}.  (87)

Indeed the filtration generated by the idle time process C̃_{t}^{n} and the filtration generated by the observation process (Ĩ_{t}^{n}, B̃_{t}^{n}) coincide, or more precisely F C̃^{n} ; t^{+} = F_{t}^{Ĩ^{n},B̃^{n}}.

Our first aim is to study the conditional law

E[g(Q̃_{t}^{n})/H̃_{t}^{n}],  (88)

where for the notational convenience we denote

H̃_{t}^{n} = F_{t}^{Ĩ^{n},B̃^{n}} = F C̃^{n} ; t^{+},

and the explicit expression for the filter (88) in terms of γ_{t}^{0}(Q̃^{n}), is given in (95). Then we consider the rescaled processes

Q_{t}^{n} := \frac{Q̃_{nt}^{n}}{\sqrt{n}} ,  I_{t}^{n} := \frac{Ĩ_{nt}^{n}}{\sqrt{n}},  B_{t}^{n} := \frac{B̃_{nt}^{n}}{\sqrt{n}} ,  C_{t}^{n} := \sqrt{n}μ_{n}C̃_{nt}^{n},

and the conditional law of the rescaled queue

E[g(Q_{t}^{n})/H_{t}^{n}],  (89)

where H_{t}^{n} is the filtration generated by the rescaled observation

H_{t}^{n} = H̃_{nt}^{n} = F_{t}^{I^{n},B^{n}} = F_{t^{+}}^{C^{n}}.

We are interested in the limit behaviour of the filter (89) under the same assumption A1, A2, A3 and conditions C1, C2, C3 of Section 5. Under these assumptions we already know that Q_{t}^{n} converge weakly to a Brownian motion W_{t} with diffusion coefficient 2λ and drift coefficient c. If one defines X̄_{t}^{n} := Q_{t}^{n} − C_{t}^{n}, then clearly Q_{t}^{n} = X̄_{t}^{n} + C_{t}^{n}, and therefore, since by definition C_{t}^{n} increases only when Q_{t}^{n} = 0, the pair (Q_{t}^{n}, C_{t}^{n}) is the solution of the Skorohod problem corresponding to X̄_{t}^{n}. Moreover

(X̄_{t}^{n}, Q_{t}^{n}, C_{t}^{n}) ⇒ (W_{t}, W_{t} + Λ_{t}, Λ_{t}) ,

where as usual Λ_{t} is the local time of W_{t} (for a deeper investigation of these results, we refer to Kurtz [8]). It is therefore natural to expect that E[g(Q_{t}^{n})/H_{t}^{n}] converges weakly to E[g(W_{t} + Λ_{t})/F_{t}^{Λ}]= Π̂_{2λ,c}(ζ_{t}; g). This result is proven in Theorem 6.4. Moreover Π̂_{2λ,c}(γ_{t}^{0}(Q^{n}); g) is a good approximation of the filter for the rescaled model (see Theorem 6.6).

Let {σ_{k}^{Bn}, k ∈ ℕ} and {σ_{k}^{In}, k ∈ ℕ} be the jump times of the process Ĩ_{t}^{n} and the process B̃_{t}^{n}, respectively. Under the assumption Q̃_{0}^{n} = 0, it easy to verify that

σ_{k}^{Bn} < σ_{k}^{In} < σ_{k+1}^{Bn} < σ_{k+1}^{In}, for each k ≥ 1,

and that Q_{t}^{n} = 0 when σ_{k}^{In} ≤ t < σ_{k+1}^{Bn}, for k ≥ 0, while Q_{t}^{n} > 0 otherwise. We start by observing some regenerative properties of the above jump times, which are fundamental in the sequel.

Lemma 6.1. For each k ∈ ℕ the processes Q̃_{k,t}^{In} = Q̃_{t+σ_{k}^{In}}^{n} − Q̃_{σ_{k}^{In}}^{n} and Q̃_{k,t}^{Bn} = Q̃_{t+σ_{k}^{Bn}}^{n} − Q̃_{σ_{k}^{Bn}}^{n}

are independent of F Q̃^{n} ; σ_{k}^{In} and F Q̃^{n} ; σ_{k}^{Bn} respectively. Moreover, the process Q̃_{k,t}^{In} has the same law as

the process Q̃_{t}^{n}.

Proof. Clearly Q̃_{k,0}^{In} = Q̃_{0}^{n} = 0, moreover it is easy to see that Q̃_{k,t}^{In} solves the same martingale problem as Q̃_{t}^{n}, hence these processes share the same law. The independence property follows by the strong Markov property of Q̃_{t}^{n}, since Q̃_{σ_{k}^{In}}^{n} = 0:

E [exp (iu (Q̃_{t+σ_{k}^{In}}^{n} − Q̃_{σ_{k}^{In}}^{n})) /F Q̃^{n} ; σ_{k}^{In}] = E [exp (iu (Q̃_{t+σ_{k}^{In}}^{n} − Q̃_{σ_{k}^{In}}^{n})) /Q̃_{σ_{k}^{In}}^{n}] =
=E [exp (iu (Q̃_{t+σ_{k}^{In}}^{n} − Q̃_{σ_{k}^{In}}^{n}))] = E [exp (iu (Q̃_{t+σ_{k}^{In}}^{n} − Q̃_{σ_{k}^{In}}^{n}))]
=E [exp (iu \binom{}{Q̃_{t}^{n}})] .

Similar arguments apply to show that the process Q̃_{k,t}^{Bn} is independent of F Q̃^{n} ; σ_{k}^{Bn}.

The process Ĩ_{t}^{n} is a renewal process, and B̃_{t}^{n} is a delayed renewal process, i.e. the random variables σ_{k+1}^{Bn} − σ_{k}^{Bn} are mutually independent for k ≥ 0 and identically distributed for k ≥ 1. Also {σ_{k}^{In} − σ_{k}^{Bn}}_{k≥1} is a sequence of mutually independent random variables. In the setting of this section, the above considerations and Lemma 6.1 guarantee that the filter of Q̃_{t}^{n} given H̃_{t}^{n} admits a representation similar to that given in Proposition 2.1. More precisely

Proposition 6.2. The conditional law of Q̃_{t}^{n} given H̃_{t}^{n} admits the following representation

E[g(Q̃_{t}^{n})/H̃_{t}^{n}] = 𝕀 (Q̃_{t}^{n} = 0) g(0)+  (90)

+𝕀 (Q̃_{t}^{n} > 0) ∑_{j=1}^{∞} \frac{E[g(Q̃_{s+σ_{j}^{Bn}}^{n} − Q̃_{σ_{j}^{Bn}}^{n} + 1)𝕀 (σ_{j}^{In} − σ_{j}^{Bn} > s) ]_{s=t−σ_{j}^{Bn}}}{E[𝕀(σ_{j}^{In} − σ_{j}^{Bn} > s)]_{s=t−σ_{j}^{Bn}}} 𝕀\{σ_{j}^{Bn} ≤ t < σ_{j}^{In}\}.

Even if this Proposition is quite similar to Proposition 2.1, we sketch the proof because the assumptions on the processes involved are a little different. Moreover we point out that since the processes involved are all Markovian, the proof of (90) could be given by the techniques used in [3].

Proof. We begin by noting that 𝕀 (Q̃_{t}^{n} > 0) is H̃_{t}^{n}-adapted. Moreover we can state (see [2] Chapter. III T5)

H̃_{t}^{n} ∩ {σ_{j}^{Bn} ≤ t < σ_{j}^{In}} ∩ {Q̃_{t}^{n} > 0} = H̃_{σ_{j}^{Bn}}^{n} ∩ {σ_{j}^{Bn} ≤ t < σ_{j}^{In}} ∩ {Q̃_{t}^{n} > 0}.

By using, as in Proposition 2.1 the arguments of Proposition 3.1 in [9], we obtain

𝕀 (Q̃_{t}^{n} > 0)E[g(Q̃_{t}^{n})/H̃_{t}^{n}]𝕀{σ_{j}^{Bn} ≤ t < σ_{j}^{In}} =
𝕀 (Q̃_{t}^{n} > 0)\frac{E[g(Q̃_{t}^{n})𝕀 (σ_{j}^{In} − σ_{j}^{Bn} > t − σ_{j}^{Bn}) /H̃_{σ_{j}^{Bn}}^{n}]}{P (σ_{j}^{In} − σ_{j}^{Bn} > t − σ_{j}^{Bn}/H̃_{σ_{j}^{Bn}}^{n})}𝕀\{σ_{j}^{Bn} ≤ t < σ_{j}^{In}\}.

Note that

{σ_{j}^{In} − σ_{j}^{Bn} > s} = { inf_{0≤u≤s} Q̃_{j,u}^{Bn} > 0}

and is independent of F Q̃^{n} ; σ_{j}^{Bn} by Lemma 6.1. Then, denoting by s = t − σ_{j}^{Bn}, so that

Q̃_{t}^{n} = Q̃_{σ_{j}^{Bn}+s}^{n} = Q̃_{σ_{j}^{Bn}+s}^{n} − Q̃_{σ_{j}^{Bn}}^{n} + 1 = Q̃_{j,s}^{Bn} + 1

\frac{E[g(Q̃_{σ_{j}^{Bn}+s}^{n})𝕀 (σ_{j}^{In} − σ_{j}^{Bn} > s) |_{s=t−σ_{j}^{Bn}}/H̃_{σ_{j}^{Bn}}^{n}]}{P ( (σ_{j}^{In} − σ_{j}^{Bn} > s) |_{s=t−σ_{j}^{Bn}}/H̃_{σ_{j}^{Bn}}^{n})}

(91)

= \frac{E[g (Q̃_{j,s}^{Bn} + 1) 𝕀 (σ_{j}^{In} − σ_{j}^{Bn} > s) ]_{s=t−σ_{j}^{Bn}}}{E[𝕀 (σ_{j}^{In} − σ_{j}^{Bn} > s) ]_{s=t−σ_{j}^{Bn}}} ,

Then the thesis follows by writing g(Q̃_{t}^{n}) as

g(Q̃_{t}^{n}) = 𝕀 (Q̃_{t}^{n} = 0) g(0) + 𝕀 (Q̃_{t}^{n} > 0) ∑_{j=1}^{∞} g(Q̃_{t}^{n})𝕀{σ_{j}^{Bn} ≤ t < σ_{j}^{In}},

and, therefore, E[g(Q̃_{t}^{n})/H̃_{t}^{n}] as

E[g(Q̃_{t}^{n})/H̃_{t}^{n}] = 𝕀 (Q̃_{t}^{n} = 0) E[g(0)/H̃_{t}^{n}] + 𝕀 (Q̃_{t}^{n} > 0) ∑_{j=1}^{∞} E[g(Q̃_{t}^{n})/H̃_{t}^{n}]𝕀{σ_{j}^{Bn} ≤ t < σ_{j}^{In}}.

It is important to note that

σ_{j}^{In} − σ_{j}^{Bn} = inf {u ≥ 0 : Q̃_{j,u}^{Bn} + 1 = 0} ,  (92)

and that the process Q̃_{j,s}^{Bn} + 1 for s < σ_{j}^{In} − σ_{j}^{Bn} behaves like the continuous time random walk X̃_{s}^{n} + 1 for s < σ̃_{1}^{n} = inf {u ≥ 0 : X̃_{u}^{n} = −1} , and hence

\frac{E[g (Q̃_{j,s}^{Bn} + 1) 𝕀 (σ_{j}^{In} − σ_{j}^{Bn} > s) ]}{E[𝕀 (σ_{j}^{In} − σ_{j}^{Bn} > s) ]} = \frac{E[g (X̃_{s}^{n} + 1) 𝕀 (σ̃_{1}^{n} > s) ]}{E[𝕀 (σ̃_{1}^{n} > s) ]}.  (93)

As a consequence

E[g(Q̃_{t}^{n})/H̃_{t}^{n}] = 𝕀 (Q̃_{t}^{n} = 0) g(0)+  (94)
+𝕀 (Q̃_{t}^{n} > 0) ∑_{j=1}^{∞} \frac{E[g (X̃_{s}^{n} + 1) 𝕀 (σ̃_{1}^{n} > s) ]_{s=t−σ_{j}^{Bn}}}{E[𝕀 (σ̃_{1}^{n} > s) ]_{s=t−σ_{j}^{Bn}}} 𝕀\{σ_{j}^{Bn} ≤ t < σ_{j}^{In}\}.

Observing that, by definition (8),

γ_{t}^{0}(Q̃^{n}) = t − sup{s < t such that Q̃_{s}^{n} = 0} = ∑_{j=1}^{∞}(t − σ_{j}^{Bn})𝕀{σ_{j}^{Bn} ≤ t < σ_{j}^{In}}

we can rewrite

E[g(Q̃_{t}^{n})/H̃_{t}^{n}] = 𝕀 (Q̃_{t}^{n} = 0) g(0) + 𝕀 (Q̃_{t}^{n} > 0) \frac{E[g (X̃_{s}^{n} + 1) 𝕀 (σ̃_{1}^{n} > s) ]}{E[𝕀 (σ̃_{1}^{n} > s) ]}|_{s=γ_{t}^{0}(Q̃^{n})} . (95)

The above considerations leads us to state the following result

Theorem 6.3. Consider the rescaled process Q_{t}^{n} = \frac{Q̃_{nt}^{n}}{\sqrt{n}}, the rescaled observation processes I_{t}^{n} = \frac{Ĩ_{nt}^{n}}{\sqrt{n}} and B_{t}^{n} = \frac{B̃_{nt}^{n}}{\sqrt{n}}, and denote by H_{t}^{n} the history generated by (I_{u}^{n}, B_{u}^{n}) for u ≤ t, i.e.

H_{t}^{n} = F_{t}^{I^{n},B^{n}} = H̃_{nt}^{n}.

Then

E[g(Q_{t}^{n})/H_{t}^{n}] = 𝕀 (Q_{t}^{n} = 0) g(0) + 𝕀 (Q_{t}^{n} > 0) Σ̄^{n}(γ_{t}^{0}(Q^{n}); g),  (96)

where

Σ̄^{n}(s; g) = Σ̂^{n}(s; \overline{g}_{n}),

with Σ̂^{n}(s) the probability measure defined in (26), and \overline{g}_{n}(x) = g(x + \frac{1}{\sqrt{n}}). Proof. Equality (95) implies

E[g(Q_{t}^{n})/H_{t}^{n}] = 𝕀 (Q_{t}^{n} = 0) g(0) + 𝕀 (Q_{t}^{n} > 0) \frac{E[g (X_{s}^{n} + \frac{1}{\sqrt{n}}) 𝕀 (σ_{1}^{n} > s) ]}{E[𝕀 (σ_{1}^{n} > s) ]}|_{s=γ_{t}^{0}(Q^{n})}

and clearly

\frac{E[g (X_{s}^{n} + \frac{1}{\sqrt{n}}) 𝕀 (σ_{1}^{n} > s) ]}{E[𝕀 (σ_{1}^{n} > s) ]} = Σ̂^{n}(s; \overline{g}_{n}).

As a consequence of the above Theorem, the conditional law of Q_{t}^{n} given H_{t}^{n} can be written as

𝕀 (Q_{t}^{n} = 0) [δ_{{0}} − Σ̄^{n}(γ_{t}^{0}(Q^{n}))] + Σ̄^{n}(γ_{t}^{0}(Q^{n})).

Moreover, for any g uniformly continuous

Σ̄^{n}(γ_{t}^{0}(Q^{n}); g) = Σ̂^{n}(γ_{t}^{0}(Q^{n}); g) + ε(n, g)  (97)

with

|ε(n, g)| ≤ ω_{g} (\frac{1}{\sqrt{n}}) = sup_{|x−y|≤ \frac{1}{\sqrt{n}}} |g(x) − g(y)|.

We can now prove the main result of this section.

Theorem 6.4. Assume conditions C1, C2, C3 and A1, A2, A3. Then, for any t ≥ 0, the sequence of measure-valued random variables defined by (89) converge weakly to π̂_{t}, on the space of probability measures endowed with the topology of weak convergence. In particular, for any bounded and continuous function g

E[g(Q_{t}^{n})/H_{t}^{n}] ⇒ E[g(W_{t})/F_{t}^{Λ}] = Π̂_{2λ,c}(ζ_{t}; g), for any t ≥ 0.

Proof. As in the proof of Theorem 5.1, using (97), it is possible to show that in the Skorohod space of Proposition 5.3

P̄(Σ̄^{n}(γ_{t}^{0}(Q^{n}); g) ⟶_{n→∞} Π̂_{2λ,c}(ζ_{t}; g), for every g : ℝ^{+} → ℝ bounded and continuous) = 1,

since in that space γ_{t}^{0}(Q^{n}) converges to ζ_{t} almost surely. On the other hand, the total variation of the measure δ_{{0}} − Σ̄^{n}(γ_{t}^{0}(Q^{n})) is at most 2, so that the result is achieved once we prove that

𝕀 (Q_{t}^{n} = 0) ⟶^{P rob} 0, for any t ≥ 0

Indeed, as recalled in (44), the sequence Q_{t}^{n} converges weakly to a reflected Brownian motion W_{t} + Λ_{t}. Then, the above convergence can be obtained by noting that

- the function 𝕀 (x = 0) has a discontinuity point at x = 0,

- P (W_{t} + Λ_{t} = 0) = 0,

- the continuous mapping theorem implies that 𝕀 (Q_{s}^{n} = 0) ⇒ 0.

Remark 6.5. As already observed at the beginning of this section,

(X̄_{t}^{n}, Q_{t}^{n}, C_{t}^{n}) ⇒ (W_{t}, W_{t} + Λ_{t}, Λ_{t})

where X̄_{t}^{n} := Q_{t}^{n} − C_{t}^{n}. Thanks to the continuity of the limit processes, the convergence can be considered in the space D_{ℝ^{3}[0,∞)} endowed with the topology of the uniform convergence on compact sets. Moreover it is interesting to note that γ_{t}^{0}(Q^{n}) = γ_{t}(C^{n}) ⇒ γ_{t}(Λ) = ζ_{t}, Then, similarly to Proposition 5.3, it is possible to prove that

(γ_{t}^{0}(Q^{n}), γ_{t}(C^{n}), C_{t}^{n}) ⇒ (γ_{t}^{0}(W_{t} + Λ_{t}), γ_{t}^{0}(Λ_{t}), Λ_{t}) = (ζ_{t}, ζ_{t}, Λ_{t}) ,

and therefore an alternative proof of the previous Theorem can be achieved, by using these properties.

We end this section by noting that, even in this new situation, it is possible to give the same approximation for the filter as in Theorem 5.2, namely for E[g(Q_{t}^{n})/H_{t}^{n}]. More precisely the following result holds

Theorem 6.6. For all g bounded and continuous and for each T > 0, p > 0

∫_{0}^{T} E|E[g(Q_{t}^{n})/H_{t}^{n}] − Π̂_{2λ,c}(γ_{t}^{0}(Q^{n}); g)|^{p}dt ⟶_{n→∞}0.

Proof. Note that, by (97),

|E[g(Q_{t}^{n})/H_{t}^{n}] − Π̂_{2λ,c}(γ_{t}^{0}(Q^{n}); g)|^{p} =
= |𝕀 (Q_{t}^{n} = 0) g(0) + 𝕀 (Q_{t}^{n} > 0) Σ̂^{n}(γ_{t}^{0}(Q^{n}); g) + ε(n, g) − Π̂_{2λ,c}(γ_{t}^{0}(Q^{n}); g)|^{p}
≤ C(p)|𝕀 (Q_{t}^{n} = 0) (g(0) + Σ̂^{n}(γ_{t}^{0}(Q^{n}); g)) + ε(n, g)|^{p}
+ C(p)|Σ̂^{n}(γ_{t}^{0}(Q^{n}); g) − Π̂_{2λ,c}(γ_{t}^{0}(Q^{n}); g)|^{p},

where C(p) is a suitable constant. Then

E [|E[g(Q_{t}^{n})/H_{t}^{n}] − Π̂_{2λ,c}(γ_{t}^{0}(Q^{n}); g)|^{p}] ≤
≤C(p)E [|𝕀 (Q_{t}^{n} = 0) 2 ∥ g ∥_{∞} +ε(n, g)|^{p}] + C(p)E [|Σ̂^{n}(γ_{t}^{0}(Q^{n}); g) − Π̂_{2λ,c}(γ_{t}^{0}(Q^{n}); g)|^{p}] .

The thesis follows since both the addends at the right hand side of the previous inequality converge to zero. The first addend converges to zero by the bounded convergence theorem. To prove that the the second addend converges to zero one has just to substitute ξ_{t}^{n} with γ_{t}^{0}(Q^{n}) in the proof of Theorem 5.2.

Acknowledgements One of us, B. Torti, started to study this kind of problems during the PhD Thesis under the guide of Anna Gerardi, as a simplified version of a problem originally posed by Thomas G. Kurtz. We thank both of them. Moreover we are indebted to Thomas G. Kurtz for the proof of Proposition 5.3.

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