The conditional law of a continuous time random walk with respect to its local time: explicit form and approximations^{∗}

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

January 2003

Abstract A filtering problem is considered in the case when the state process is a continuous time random walk X_{t} and the observation process is its local time L_{t}. An explicit construction of the filter is given. This construction is then applied to a suitable approximation of a Brownian motion and to an asymptotically symmetric M/M/1 queueing model. In both these cases it has been possible to give a convergence result for the respective filters.

Contents

1 Introduction 2 2 The model 4 3 Scaling 8 4 The interpolating Brownian motion model: strong convergence for the filter 10 5 The M/M/1 queueing model: weak convergence for the filter 14

5.1 Weak convergence of the filter: the symmetric case . . . . . . . . . . . . . . . . . 17 5.2 Weak convergence of the filter: the asymptotically symmetric case . . . . . . . . 21

6 The M/M/1 queueing model: an approximation for the filter 24 7 The M/M/1 queueing model: observing the idle time process 26 A Appendix 33

A.1 Identification of the interpolating Brownian motion model . . . . . . . . . . . . . 33 B A limit result and an upper bound for the rescaled queue 35

B.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 B.2 Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 B.3 Technical details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

∗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 [14]). 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} = − inf_{0≤s≤t} (W_{s}) is the corresponding local time in the sense of Skorohod definition. In the limit model the 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 filter of the limit model of reflected Brownian motion, and for the filter of 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 [13] (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 [13]. 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(g)(s, l) = ∫_{0}^{∞} g (−l + y\sqrt{s}) y exp (−\frac{y^{2}}{2} ) dy s ≥ 0,  (1)

or, equivalently,

Π(g)(s, l) = E[g(−l + W_{s}^{∗})/Λ_{s}^{∗} = 0],  (2)

where W^{∗} is any standard Brownian motion and Λ^{∗} is its local time.

Then

π_{t}(g) = E[g(W_{t})/F_{t}^{Λ}] = \frac{Π (g(·) exp(\frac{c}{a^{2}} ·)) (a^{2}s, l)}{Π (exp(\frac{c}{a^{2}} ·)) (a^{2}s, l)}|_{s=ζ_{t},l=Λ_{t}} ,  (3)

and

π̂_{t}(g) = E[g(W_{t} + Λ_{t})/F_{t}^{Λ}] = \frac{Π (g(·) exp(\frac{c}{a^{2}} ·)) (a^{2}s, 0)}{Π (exp(\frac{c}{a^{2}} ·)) (a^{2}s, 0)}|_{s=ζ_{t}} ,  (4)

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),  (5)

where

γ_{t}^{0}(x) = t − sup{s < t : x_{s} = 0}.  (6)

γ_{t}(x) = t − sup{s < t : x_{s} < x_{t}}.  (7)

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} up to time t. Indeed, up to a reflection, several queueing models and birth and death processes can be represented just by choosing the law of a continuous time random walk.

More precisely, we start by deriving an 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. It turns out that the filter of Y_{t} w.r.t. L_{u} up to time t can be expressed as a deterministic function of L_{t} and ζ_{t} = γ_{t}(L), where γ_{t} is the deterministic functional of L_{u} up to time t defined in (7).

Then, we consider two situations when a sequence (X_{t}^{n}, L_{t}^{n}), of rescaled random walks, converges to a Brownian motion and its local time (W_{t}, Λ_{t}), and we deal with the problem whether the corresponding filter converges to the filter of the limit. As discussed at the end of Section 3, the convergence of the previous systems is equivalent to the convergence of the system with the reflected random walk (X_{t}^{n} + L_{t}^{n}, L_{t}^{n}) to the system with the reflected Brownian motion (W_{t} + Λ_{t}, Λ_{t}), and the convergence of the filters for the rescaled random walks is equivalent to the convergence of the filters for the reflected random walks.

In particular, for the models we propose, we study two different kind of problems. The first is a theoretical one, and concerns the problem of checking if the weak limit of the filter is the the filter of the limit model. In turn, this question give rise to another problem, more interesting from a computational point of view, regarding to find a good approximation of the filter of the discrete system. If the filter, as functional of the observation process, satisfies some regularity properties both in the discrete case and in the limit case, we can think to use the limit functional, computed on the true observation, in order to give a manageable approximation of the filter. This situation will be analyzed in Section 6. 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 ([9], [10]). 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 in the sense specified above. The above problem of 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 [9]). 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 the example above, 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 [9] (proof of Theorem 2.1, Step 1): one has only to replace the sequence of σ−algebras used in [9] 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 [10] to study a filter approximation problem in diffusive models, and is related with the approximation scheme used in [13]. In addition the processes W^{n} can be identified with our rescaled model. In this particular case we also get a strong convergence result for the approximating filter.

The second model we consider is the case when the renewal process Z_{t} is a Poisson process, so that the reflected random walk is an M/M/1 queue. We consider the case when the drift of the limit Brownian motion is zero, and we find, under suitable conditions, the weak limit of the corresponding filter by a limiting procedure. It is important to note that the observation process is the local time of the random walk, which is not the cumulative time spent in 0, as in the original motivating problem. The last Section outlines how this problem can be dealt with.

2 The model

Fix a probability space (Ω, F, P) and consider on it two sequences of random variables {T_{j}, j ≥ 1}, {U_{j}, j ≥ 1} satisfying the assumptions

H1 the sequences {T_{j}, j ≥ 1} {U_{j}, j ≥ 1} are mutually independent;

H2 the random variables {T_{j}, j ≥ 1} are non-negative, mutually independent, with identical distribution function F;

H3 {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).

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 marked point process

Y_{t} = V_{Z_{t}} = ∑_{j=1}^{∞} U_{j}𝕀 (τ_{j} ≤ t) = ∑_{j≤Z_{t}} U_{j}.  (8)

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 : D_{ℝ}[0, ∞) → D_{ℝ}[0, ∞) s.t. ℓ_{t}(x) = − inf_{s≤t} x(s) ∧ 0.  (9)

It is easy to see that, for our model, L_{t} admits the representation

L_{t} = ∑_{j=1}^{∞} 𝕀 (σ_{j} ≤ t) ,  (10)

where {σ_{j}, j ≥ 0} is the sequence of its jump times. We can see that {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.

Set G_{t} = F_{t}^{L} = σ {L_{s}, s ≤ t} and F_{t} = F_{t}^{Y} = σ {Y_{s}, s ≤ t}. Obviously {σ_{j}, j ≥ 0} are stopping times w.r.t. both the histories G_{t} and F_{t}.

A first representation for the filter of Y_{t} given G_{t} is stated in the following Proposition, where we use a condition, which is implied by the above conditions H1, H2, H3.

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

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}\},  (11)

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 [13], 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 (11) admits the representation

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.

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}\},  (12)

where Y_{s}^{∗} is another birth and death process, with the same law as Y_{t}, defined by the rule (8) 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

\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)]},

for any s ≥ 0.

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}  (13)

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 (and therefore under Conditions H2 and H3 for a suitable F and a suitable value of p) the sequence {M_{i}, i ≥ 1} is a sequence of i.i.d. random variables. Moreover, under Condition H1, it is obvious that {M_{i}, i ≥ 1} and {T_{i}, i ≥ 1} are mutually independent. Next result provides an explicit expression for the terms of the sum in (11).

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}\} (14)

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

Finally, taking into account also (8),

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

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

= ∑_{m=1}^{∞} E[𝕀 (M_{1} = m) ∑_{i=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)).

Now the state space of the process Y_{t} is discrete and then 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. In the next Proposition we give the expression of the discrete density ν_{t}(x).

Proposition 2.4. The filter of Y_{t} given G_{t} has discrete density

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}  (15)

Proof. By using (14) for the functions g(z) = 𝕀_{{x}}(z), z ∈ ℤ we achieve the result.

The joint law of (V_{k−1}, max_{j≤k−1} V_{j}) is well-known (see e.g. Theorem 3.10.10 in [11]), and in particular

P (V_{k−1} = −a, max_{j≤k−1} V_{j} < 1) =

= \begin{cases}\frac{a + 1}{k}\binom{k}{\frac{k+a+1}{2}}q^{\frac{k+a−1}{2}} p^{\frac{k−a−1}{2}} & if k + a − 1 is even and |a| ≤ k − 1 \\ 0 & otherwise. \end{cases}  (16)

Observe that {V_{k−1} = a, M_{1} ≥ k} = {V_{k−1} = a, min_{j≤k−1} V_{j} > −1}. Then by noting that

P (V_{k−1} = −a, max_{j≤k−1} V_{j} < 1) = P (−V_{k−1} = a, min_{j≤k−1}(−V_{j}) > −1)

we obtain the thesis interchanging the role of p and q in (16).

3 Scaling

Let {X̃_{t}^{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 1 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^{n}[g(X_{t}^{n})/F_{t}^{L^{n}}] = E^{n}[g(b_{n}X̃_{a_{n}t}^{n})/F L̃^{n} ; a_{n}t],

where E^{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 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) = E[g(X_{t}^{n})/G_{t}^{n}] = ∑_{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}) , (17)

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};

• 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}}.

This observation allows us to write (17) as follows

π_{t}^{n}(g) = E[g(X_{t}^{n})/G_{t}^{n}] = E[g(−l + X_{s}^{n∗})/{L_{s}^{n∗} < b_{n}}]|_{l=L_{t}^{n}, s=ξ_{t}^{n}} =
= E[g(−l + X_{s}^{n∗})/{L_{s}^{n∗} = 0}]|_{l=L_{t}^{n}, s=ξ_{t}^{n}},  (18)

where

ξ_{t}^{n} = t − sup{u ≤ t s. t. L_{u}^{n} < L_{t}^{n}} = γ_{t}(L^{n}),  (19)

with γ_{t}(x) the functional defined by (7). Then, denoting by Σ^{n}(g)(s, l)

Σ^{n}(g)(s, l) = E[g(−l + X_{s}^{n∗})/\{L_{s}^{n∗} < b_{n}\}] = \frac{E[g(X_{s}^{n∗} − l)𝕀 (σ_{1}^{n∗} > s) ]}{E[𝕀(σ_{1}^{n∗} > s)]}  (20)

(17) can be shortly written as

π_{t}^{n}(g) = E[g(X_{t}^{n})/G_{t}^{n}] = Σ^{n}(g)(ξ_{t}^{n}, L_{t}^{n}).  (21)

Remark 3.1. Note that Σ^{n}(g)(s, l) depends also on the probability measure P^{n}, i.e. Σ^{n}(g)(s, l) = Σ_{P^{n}}^{n}(g)(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}(g)(ξ_{t}^{n}),  (22)

where

Σ̂^{n}(g)(s) = Σ^{n}(g)(s, 0) = E [g(X_{s}^{n∗})/\{L_{s}^{n∗} < b_{n}\}] = \frac{E[g(X_{s}^{n∗})𝕀 (σ_{1}^{n∗} > s) ]}{E[𝕀(σ_{1}^{n∗} > s)]}.  (23)

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

Σ^{n}(g)(s, l) = \frac{∑_{k=1}^{∞} E[𝕀 (M_{1}^{n} ≥ k) g(b_{n}V_{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))}  (24)

Σ̂^{n}(g)(s) = \frac{∑_{k=1}^{∞} E[𝕀 (M_{1}^{n} ≥ k) g(b_{n}V_{k−1}^{n})](F_{k−1}^{n}(s) − F_{k}^{n}(s))}{∑_{m=1}^{∞} P (M_{1}^{n} = m)(1 − F_{m}^{n}(s))},  (25)

where M_{1}^{n}, F_{k}^{n} have a similar meaning as M_{1}, F_{k} in (14). 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 remainder of this paper is devoted to the problem of finding the limit of the filter (21) ( or of the filter (22) ), or some good approximation for it, in the case when the rescaled sequence X_{t}^{n} converges to a process X_{t}. When X_{t}^{n} converges to a continuous process X_{t}, then the local time L_{t}^{n} = ℓ_{t}(X^{n}) of X_{t}^{n} also converges to the local time ℓ_{t}(X) of X_{t}. In fact the functional ℓ_{t}(x) defined by (9) is continuous w.r.t. the topology of the uniform convergence on compact sets. In this case it is also natural to investigate whether the limit of the filter is the corresponding filter of the limits. Finally we observe that, if the limit of (22) exists, then the same holds for (21), and these limits can be obtained one from each other, as we can obtain (22) by (21) and vice-versa.

4 The interpolating Brownian motion model: strong convergence for the filter

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. When the process W has drift zero, then the approximating model falls into the frame of the previous sections, in particular the common distribution function of T_{j} is F^{0} (see (39) below) and the random walk is symmetric. The asymmetric case can also be managed, and we will briefly explain how at the end of this Section.

We start by introducing the approximating model, then we show the convergence result.

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}  (26)

Put z_{k}^{h} = z(τ_{k}^{h}(z)) and consider the function z^{h} ∈ D_{ℝ}[0, +∞)

z^{h}(t) = ∑_{k=0}^{∞} 𝕀_{[τ_{k}^{h}(z),τ_{k+1}^{h}(z))}(t)z_{k}^{h}.  (27)

We will need the following results

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

|z^{h}(t) − z(t)| ≤ h,  (28)

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 (9).

Proof. The bound (28) is due to the continuity of z, and the uniform convergence of z^{h} to z follows. The continuity of the functional ℓ implies the other limit.

Remark 4.2. 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}.  (29)

When z is a continuous function and 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)  (30)

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 (9).

Observe that W_{0} = 0 implies

Λ_{t} = − inf_{0≤s≤t} (W_{s}).  (31)

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 (26). Then the approximating signal/observation process (W^{n}, ℓ(W^{n})) is a D_{ℝ^{2}}[0, +∞)-valued process, where

W_{t}^{n} = ∑_{k=0}^{∞} W_{k}^{n} 𝕀_{[τ_{k}^{n},τ_{k+1}^{n})}(t), with W_{k}^{n} = W (τ_{k}^{n}).  (32)

Note that Lemma 4.1 provides the following convergence result.

Lemma 4.3. Let W^{n} be defined as in (32). Then, for each t ∈ ℝ^{+}

|W_{t}^{n} − W_{t}| ≤ \frac{1}{2^{n}} ,  (33)

and

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

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

Remark 4.2 provides

ℓ_{t}(W^{n}) = ∑_{j=0}^{∞} \frac{1}{2^{n}} 𝕀(σ_{j}^{n} ≤ t),  (34)

where

σ_{j}^{n} = inf \{t s.t. W_{t} ≤ −\frac{j}{2^{n}} \} = inf \{t s.t. Λ_{t} ≥ \frac{j}{2^{n}} \}.  (35)

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 {G_{t}^{n} = σ(ℓ_{s}(W^{n}), s ≤ t), n ∈ ℕ} is an increasing family of σ-algebras, and allows us to show the claimed strong convergence result.

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

E[g(W_{t}^{n})/G_{t}^{n}] → E[g(W_{t})/F_{t}^{Λ}], a.s. and in L^{1},  (36)

where E[g(W_{t})/F_{t}^{Λ}] is given in (3).

Proof. Observe that

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

In [13] the r.h.s. of (34) defines the process Λ_{t}^{n}, and then ℓ_{t}(W^{n}) = Λ_{t}^{n}, and G_{t}^{n} = σ(Λ_{s}^{n}, s ≤ t). Moreover G_{t}^{n} ↑ F_{t}^{Λ} (see Lemma 2.3 of [13]) and then we get (see Theorem 2.4 of [13])

E[g(W_{t})/G_{t}^{n}] → E[g(W_{t})/F_{t}^{Λ}]  a.s. and in L^{1},  (37)

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

E|E[g(W_{t})/G_{t}^{n}] − E[g(W_{t}^{n})/G_{t}^{n}]| ≤ E|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 (36) states that the conditional laws defined by E[g(W_{t}^{n})/G_{t}^{n}] converge a.s. to the conditional law defined by (3).

By Lemma 4.3, we are in the situation discussed at the end of Section 3 with X^{n} = W^{n} and X = W. Then one can get the analogous convergence results for the corresponding queueing model generated by reflecting W^{n}. In particular the conditional laws E[g(W_{t}^{n} + ℓ_{t}(W^{n}))/G_{t}^{n}] converge a.s. to E[g(W_{t} + Λ_{t})/F_{t}^{Λ}], (see (4)).

We now start to show that the approximating model falls into the frame of the previous sections. When the drift is zero, we can assume without loss of generality that W is a standard Brownian motion (when the diffusion coefficient is a^{2} we can use the deterministic time change \frac{t}{a^{2}} instead of t). In this case the process W_{t}^{n} can be represented on the space (Ω, F, P) as follows

W_{t}^{n} = \frac{1}{2^{n}} V_{Z_{t}^{n}}^{n},  (38)

where

• Z_{t}^{n} is a renewal process defined by the sequence of i.i.d. interarrival times {T_{k}^{n} = τ_{k}^{n} − τ_{k−1}^{n}, k ∈ ℕ} with common law (see for example [7] page 342)

F^{n}(t) = P (T_{k}^{n} ≤ t) = 4 ∑_{j=0}^{+∞}(−1)^{j} \frac{1}{\sqrt{2π}} ∫ +∞_{\frac{(2j+1)}{2^{n}\sqrt{t}}} exp (−\frac{x^{2}}{2} ) dx;  (39)

• V_{k}^{n} = 2^{n}W_{τ_{k}^{n}} is a symmetric random walk;

• Z_{t}^{n} and V_{k}^{n} are mutually independent.

Let V be the symmetric random walk V^{0}, and Z be the renewal process Z^{0}, with interarrival time distribution F^{0}(t) (see (39) with n = 0). Define X̃_{t} = V_{Z_{t}}, and the rescaled process X_{t}^{n} = \frac{1}{2^{n}} V_{Z_{2^{2n}t}}. Then X̃ = W^{0}, and it is easy to verify that

X^{n} = W^{n} in law.

Consequently (X^{n}, ℓ(X^{n})) = (W^{n}, ℓ(W^{n})) in law, where ℓ is the functional defined by (9). Therefore, recalling (21), E[g(W_{t}^{n})/G_{t}^{n}] = Σ^{n}(g)(γ_{t}(W^{n}), ℓ(W^{n})) where Σ^{n}(g) is defined in (24), with F^{n} as in (39), and γ_{t} as in (7). A similar result holds also for E[g(W_{t}^{n} +ℓ_{t}(W^{n}))/G_{t}^{n}].

When the drift coefficient is c ≠ 0, and the diffusion coefficient is a^{2}, the approximating model does not fall into the frame of the previous sections, since Z_{t}^{n} and V_{k}^{n}, defined as above, are not mutually independent. Nevertheless condition H0 holds , and one could represent the approximating filter as explained in Remark 2.2. Alternatively 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. From this formula one can compute exactly E[g(W_{t}^{n})/G_{t}^{n}]. Moreover, for g a uniformly continuous function with modulus of continuity ω_{g}, the filter E[g(W_{t}^{n})/G_{t}^{n}] can be expressed as

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

where, taking into account that |W_{t} − W_{t}^{n}| ≤ 1/2^{n} (see Lemma 4.3),

|R(n, t)| ≤ \frac{E^{P_{0}}[|g(W_{t}) − 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}]} ≤ ω_{g}(1/2^{n}).

Therefore the filter converges a.s. to

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

5 The M/M/1 queueing model: weak convergence for the filter

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, we will get also the weak limit of the filter of the rescaled system.

In order to do so, in Subsection 5.1 we show the claimed convergence in the case when the arrival intensity λ_{n} and the service potential μ_{n} of the M/M/1 queue are equal; then, in Subsection 5.2, by using a suitable change of measure, we will extend this result to the sequences of M/M/1 queues satisfying conditions C1, C2, C3 below.

Let {λ_{n}, n ∈ ℕ} and {μ_{n}, n ∈ ℕ} be two sequences of real numbers satisfying the conditions

C1 0 < λ_{n} ≤ μ_{n}

C2 (λ_{n}, μ_{n}) →_{n→+∞} (λ, λ)

C3 \sqrt{n}(λ_{n} − λ) →_{n→+∞} 0, and \sqrt{n}(μ_{n} − λ) →_{n→+∞} 0

As a consequence of condition C3 we have \sqrt{n}(μ_{n} − λ_{n}) →_{n→+∞} 0.

The sequence of random walks we consider is defined by means of the same rule (8), namely for each n ∈ ℕ

X̃_{t}^{n} = ṼZ̃_{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}

{Ũ_{j}^{n}, j ∈ ℕ} is a sequence of i.i.d. random variables with

\begin{cases}P^{n}(Ũ_{k}^{n} = +1) = \frac{λ_{n}}{λ_{n} + μ_{n}} \\ P^{n}(Ũ_{k}^{n} = −1) = \frac{μ_{n}}{λ_{n} + μ_{n}} \end{cases}

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

So, 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}). 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

Ã_{t}^{n} = ∑_{k=0}^{∞} 𝕀(U_{k}^{n} = 1)𝕀(τ_{k}^{n} ≤ t)  (40)

Ñ_{t}^{n} = ∑_{k=0}^{∞} 𝕀(U_{k}^{n} = −1)𝕀(τ_{k}^{n} ≤ t).  (41)

Lemma 5.1. The processes Ã^{n} and Ñ^{n}, defined by (40) and (41), are mutually independent Poisson processes with intensities λ_{n} and μ_{n} respectively.

Proof. We just note that Ã^{n} and Ñ^{n} do not have common jumps and that they are Poisson processes with parameters λ_{n}, μ_{n} respectively. Then Watanabe’s Theorem (see, for instance, [3]) yields the thesis.

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, [3]). Under the assumptions stated in this Section, 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}),  (42)

where W_{t} is a Brownian motion with diffusion coefficient 2λ and Λ_{t} is the local time of W_{t}. Therefore we are in the situation discussed at the end of Section 3, with a_{n} = n and b_{n} = \frac{1}{\sqrt{n}}. In the sequel we provide the weak limit of the filter of X_{t}^{n} and Q_{t}^{n} given the filtration G_{t}^{n}, where, as in Section 3, G_{t}^{n} denotes the filtration generated by L_{t}^{n}. The main result is stated in the following Theorem which is proved in Subsection 5.2.

Theorem 5.2. Assume conditions C1, C2, C3 and A1, A2, A3, and let g be a bounded continuous function.Then

π_{t}^{n}(g) = E[g(X_{t}^{n})/G_{t}^{n}] ⇒ π_{t}(g) = E[g(W_{t})/F_{t}^{Λ}]  (43)

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}],  (44)

moreover 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.

The proof of Theorem 5.2 is based on the representations (21) and (22) 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 (6) and (7) 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 7.

Proposition 5.3. 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

sup_{s≤t}(|X_{s}^{n} − W_{s}| + |Q_{s}^{n} − W_{s} − Λ_{s}|) → 0 a.s.  (45)

This implies that L^{n} → Λ uniformly in [0, t] 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 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 (45), 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, a.s.

and then

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

The proof is achieved since

P (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 P {inf_{s≤r} W_{s} = inf_{r≤s≤t} W_{s}} = 0, and hence

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

Remark 5.4. In the Skorohod space 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 dP × 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 dP × dt-measure set.

The following remarks summarize the results we need to find the weak limit of E[g(Q_{t}^{n})/G_{t}^{n}].

• Both the filter E[g(Q_{t}^{n})/G_{t}^{n}] and the filter E[g(W_{t} + Λ_{t})/F_{t}^{Λ}] for the limit model are given as the compositions of deterministic functionals with processes measurable w.r.t. the respective observed histories. More specifically, the representation (22) for the filter E[g(Q_{t}^{n})/G_{t}^{n}] is the deterministic functional Σ̂^{n}(g)(s) (23) evaluated in s = ξ_{t}^{n} and E[g(W_{t} + Λ_{t})/F_{t}^{Λ}] is the functional Π(g)(s, 0) evaluated in s = ζ_{t} (see Theorem 1.1).

• The sequence Q^{n} converges weakly to a reflected Brownian motion, hence by Lemma 1.2, the sequence E[g(Q_{t}^{n})/G_{t}^{n}] is tight, and moreover, by Proposition 5.3, ξ_{t}^{n} ⇒ ζ_{t}.

The previous remarks induce to conjecture that the limit of E[g(Q_{t}^{n})/G_{t}^{n}] is E[g(W_{t} + Λ_{t})/F_{t}^{Λ}]. We prove this intuitive idea by developing the following steps.

• In Proposition 5.5, under the assumption λ_{n} = μ_{n} = λ, we compute the limit of the sequence of measures defined by Σ̂^{n}(g)(s) in (23), and we note that it turns out to be the probability measure defined by the functional Π(g)(s, 0) (see (2) in Theorem 1.1).

• Then, by using Proposition 5.3 we show that the weak limit of E[g(Q_{t}^{n})/G_{t}^{n}] = Σ̂^{n}(g)(ξ_{t}^{n}) is Π(g)(s, 0) evaluated in s = ζ_{t}, that is the filter of a reflected Brownian motion w.r.t. its local time (see Theorem 1.1).

• Finally we extend previous result to the more general case defined by conditions C1, C2, C3 by making a suitable change of measure. Then Theorem 5.2 is completely shown.

5.1 Weak convergence of the filter: the symmetric case

On a suitable space (Ω, F, P) let Z_{t} be a Poisson process with intensity 2 λ and let V_{k} be a symmetric random walk. Let X̃_{t} be the process defined by the rule (8), that is X̃_{t} = V_{Z_{t}}. In this particular situation the process X_{t}^{n} = \frac{X̃_{nt}}{\sqrt{n}} depends on the index n just by the time-space scaling.

These properties for the model are equivalent to assume

λ_{n} = μ_{n} = λ  (46)

and, consequently, c = 0. In this case the process W_{t} defined by (42) 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 \frac{t}{2λ} instead of t.

Denote by Π̂(s) the probability measure defined by

Π̂(g)(s) = \begin{cases}Π(g)(s, 0) = ∫_{0}^{∞} g(y\sqrt{s})y exp (−\frac{y^{2}}{2} ) dy if s > 0, \\ g(0) if s = 0, \end{cases}  (47)

for each g : ℝ^{+} → ℝ bounded and continuous, where Π(g)(s, l) is the function defined by (1) and (2) in Theorem 1.1.

Proposition 5.5. Let g : ℝ^{+} → ℝ be a bounded continuous function. Under the assumption (46), for each s,

Σ̂^{n}(g)(s) ⟶_{n→∞} Π̂(g)(s).  (48)

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

Σ̂^{n}(g)(s_{n}) ⟶_{n→∞} Π̂(g)(s).  (49)

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},  (50)

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 [7], 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}.  (51)

where F̂_{s}^{n}(x) = F_{s}^{n}(x\sqrt{s}) is the distribution function of \frac{X_{s}^{n}}{\sqrt{s}}. The expansion (51) does not imply immediately the thesis, nevertheless (50) 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}.  (52)

Proof of Proposition 5.5. We prove in detail only (48). The proof of (49) 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 (48) is trivial. When s > 0 we note that Σ̂^{n}(s) is the conditional law of X_{s}^{n∗} given the event \{L_{s}^{n∗} < \frac{1}{\sqrt{n}}\}. Then

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

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

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

By an explicit computation we get

Θ^{n}(g)(s) = ∑_{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}(g)(s) = ∑_{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\}] .  (53)

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

Θ^{n}(g)(s) = 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}(g)(s) = E [g̃′(X_{s}^{n∗} − θ \frac{2}{\sqrt{n}})] \frac{2}{\sqrt{n}} \frac{1}{Θ^{n}(1)(s)},  (54)

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})] ,  (55)

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(g)(s).

Then, by (54), the proof of (48) is completely achieved by showing that

lim_{n→∞} \frac{\sqrt{n}Θ^{n}(1)(s)}{2} = \frac{1}{\sqrt{2πs}} .  (56)

By (53) we get

Θ^{n}(1)(s) = P (X_{s}^{n∗} ∈ [0, \frac{1}{\sqrt{n}}]) = F_{s}^{n} (\frac{1}{\sqrt{n}}) − F_{s}^{n} (−\frac{1}{\sqrt{n}}) ,  (57)

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 (50), so that

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

□

Theorem 5.8. Under the same assumptions of Proposition 5.5 for each t > 0

Σ̂^{n}(g)(ξ_{t}^{n}) ⇒ Π̂(g)(ζ_{t}).  (58)

for each g : ℝ^{+} → ℝ bounded and continuous. Moreover π̂_{t}^{n} converge weakly to π̂_{t}.

Proof. Proposition 5.5 guarantees that, for any sequence s_{n} → s, s > 0,

Σ̂^{n}(g)(s_{n}) → ∫_{0}^{∞} g(\sqrt{s}x)x exp (−\frac{x^{2}}{2} ) dx = Π̂(g)(s).

Then the thesis follows since we can use the Skorohod representation probability space as in Proposition 5.3, and in this space, for each t > 0, ξ_{t}^{n} = γ_{t}(L^{n}) → ζ_{t}, a.s., and on the other hand P {ω : ζ_{t}(ω) = 0} = 0. Indeed therefore

Σ̂^{n}(g)(ξ_{t}^{n}) → Π̂(g)(ζ_{t}), for each g : ℝ^{+} → ℝ bounded and continuous a.s.  (59)

Remark 5.9. Proposition 5.5 applies also to show that, for any sequence s_{n} → s, s > 0 and for each g : ℝ^{+} → ℝ bounded continuous,

Σ^{n}(g)(s_{n}, l) ⟶_{n→∞} Π(g)(s, l) = ∫_{0}^{∞} g(−l + \sqrt{s}x)x exp (−\frac{x^{2}}{2} ) dx.  (60)

Then, following the same procedure as in Theorem 5.8 we can state that

Σ^{n}(g)(ξ_{t}^{n}, L_{t}^{n}) ⇒ Π(g)(ζ_{t}, Λ_{t}),  (61)

for each g : ℝ^{+} → ℝ bounded and continuous, and that π_{t}^{n} converge weakly to π_{t}, so that Theorem 5.2 is completely achieved in the symmetric case.

5.2 Weak convergence of the filter: the asymptotically symmetric case

In this subsection our aim is to prove Theorem 5.2 in the asymptotically symmetric case defined by conditions C1, C2, C3 and A1, A2, A3, with the same space-time scaling. The symmetric case investigated in previous subsection is obtained by defining on a suitable space (Ω, F, P) a Poisson process Z_{t} of intensity 2λ and a symmetric random walk V_{k} = ∑_{j=1}^{k} U_{j}. Starting from these processes, and in analogy with (40) and (41), 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 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} = L_{t}^{A,n} × L_{t}^{N,n}  (62)

where

L_{t}^{A,n} := exp \{∫_{0}^{t} log \frac{λ_{n}}{λ} dÂ_{s}^{n} − ∫_{0}^{t} n (λ_{n} − λ) ds\} = exp \{log (\frac{λ_{n}}{λ} ) Â_{t}^{n} − n (λ_{n} − λ) t\} (63)
L_{t}^{N,n} := exp \{∫_{0}^{t} log \frac{μ_{n}}{λ} dN̂_{s}^{n} − ∫_{0}^{t} n (μ_{n} − λ) ds\} = exp \{log (\frac{λ_{n}}{λ} ) N̂_{t}^{n} − n (λ_{n} − λ) t\} . (64)

Under the measure P, the processes Â_{t}^{n}, N̂_{t}^{n} are mutually independent Poisson processes with intensities nλ, nλ while (see [3], 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, under P^{n}, the conditions A1, A2, A3 are satisfied with Z^{n} = Z, U_{j}^{n} = U_{j}, namely:

1 Z_{t}^{n} = Â_{t}^{n} + N̂_{t}^{n} = Z_{nt} is a Poisson process with intensity n(λ_{n} + μ_{n});

2 Z_{nt}^{n} = Z_{nt} is a Poisson process with intensity n(λ_{n} + μ_{n});

3 {U_{k}, k ∈ ℕ} is a sequence of i.i.d. random variable with law

\begin{cases}P^{n}(U_{k} = 1) = & \frac{λ_{n}}{λ_{n}+μ_{n}} \\ P^{n}(U_{k} = −1) = & \frac{μ_{n}}{λ_{n}+μ_{n}} \end{cases}

4 {U_{k}, k ∈ ℕ} and Z_{t}^{n} are mutually independent.

We can now state the main result of this subsection.

Theorem 5.10. Let G_{t}^{n} be the filtration generated by L_{t}^{n}, and g be a bounded and continuous function. Then

E^{n}[g(Q_{t}^{n})/G_{t}^{n}] ⇒ Π̂(g)(ζ_{t}),  (65)

where Π̂(s) is defined by (47).

Proof. The thesis is achieved if we prove that for any bounded and Lipschitz continuous function φ

E^{n}[φ(E^{n}[g(Q_{t}^{n})/G_{t}^{n}])] → E[φ(Π̂(g)(ζ_{t}))]

|E^{n}[φ(E^{n}[g(Q_{t}^{n})/G_{t}^{n}])] − E[φ(Π̂(g)(ζ_{t}))]|
= |E[L_{t}^{n}φ(E^{n}[g(Q_{t}^{n})/G_{t}^{n}])] − E[φ(Π̂(g)(ζ_{t}))]|

≤ |E[L_{t}^{n}φ(E^{n}[g(Q_{t}^{n})/G_{t}^{n}])] − E[φ(E^{n}[g(Q_{t}^{n})/G_{t}^{n}])]|
+ |E[φ(E^{n}[g(Q_{t}^{n})/G_{t}^{n}])] − E[φ(E[g(Q_{t}^{n})/G_{t}^{n}])]|
+ |E[φ(E[g(Q_{t}^{n})/G_{t}^{n}])] − E[φ(Π̂(g)(ζ_{t}))]|

The last addend in the above inequality converges to zero by Theorem 5.8 for the symmetric case. The first one is bounded above by

|E[L_{t}^{n}φ(E^{n}[g(Q_{t}^{n})/G_{t}^{n}])] − E[φ(E^{n}[g(Q_{t}^{n})/G_{t}^{n}])]| ≤∥ φ ∥_{∞} E[ |L_{t}^{n} − 1| ].

The second addend is bounded above by

E[ |φ(E^{n}[g(Q_{t}^{n})/G_{t}^{n}]) − φ(E[g(Q_{t}^{n})/G_{t}^{n}])| ]
≤ L_{φ}E[ |E^{n}[g(Q_{t}^{n})/G_{t}^{n}] − E[g(Q_{t}^{n})/G_{t}^{n}]| ] ≤ L_{φ}2 ∥ g ∥_{∞} E[ |L_{t}^{n} − 1| ],

where L_{φ} is the Lipschitz constant of φ, and the last inequality follows by Lemma 5.12 below. The thesis is achieved since E[ |L_{t}^{n} − 1| ] converges to zero, as is proven il Lemma 5.11 below.

Lemma 5.11. Let L_{t}^{n} := \frac{dP^{n}}{dP} |_{F_{t}^{n}} be defined as in (62), then

E [(L_{t}^{n})^{2}] = exp (\frac{t}{λ} [(\sqrt{n}(λ_{n} − λ))^{2} + (\sqrt{n}(μ_{n} − λ))^{2}])  (66)

Moreover, under the conditions C1, C2, C3, on the space (Ω, F, P)

E[ |L_{t}^{n} − 1| ] → 0,  (67)

and

sup_{n} E [(L_{t}^{n})^{2}] < ∞,  (68)

Proof. We only need to prove (66), since

E^{2}[ |L_{t}^{n} − 1| ] ≤ E[ |L_{t}^{n} − 1|^{2} ]

= E[(L_{t}^{n})^{2}] − 2E[L_{t}^{n}] + 1 = E[(L_{t}^{n})^{2}] − 1

= exp (\frac{t}{λ} [(\sqrt{n}(λ_{n} − λ))^{2} + (\sqrt{n}(μ_{n} − λ))^{2}]) − 1 → 0,

and then, by condition C3, we get (67) and (68).

To this end we observe that

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

and so

E [(L_{t}^{n})^{2}] = ∑_{k=0}^{∞} (\frac{λ_{n}}{λ} )^{2k} \frac{(nλt)^{k}}{k!} exp (−nλt) exp (−2n (λ_{n} − λ) t) ×

× ∑_{k=0}^{∞} (\frac{μ_{n}}{λ} )^{2k} \frac{(nλt)^{k}}{k!} exp (−nλt) exp (−2n (μ_{n} − λ) t) =

= exp (\frac{t}{λ} [(\sqrt{n}(λ_{n} − λ))^{2} + (\sqrt{n}(μ_{n} − λ))^{2}]) .

Lemma 5.12. Let (Ω, F, ℙ) and (Ω, F, ℙ^{0}) be probability spaces with ℙ and ℙ^{0} equivalent probability measures, G ⊂ F, P = ℙ|_{G} and P^{0} = ℙ^{0}|_{G}, then

𝔼^{0} [|𝔼^{0}[ψ/G] − 𝔼[ψ/G]|] ≤ 2α𝔼^{0} [|1 − (dℙ/dℙ^{0})|] = 2α𝔼 [|(dℙ^{0}/dℙ) − 1|] ,

for any random variable ψ bounded above by α = ∥ψ∥_{∞}. The same bound holds for 𝔼 [|𝔼^{0}[ψ/G] − 𝔼[ψ/G]|].

Proof. The proof is based on the Kallianpur-Striebel formula and is a particular case of Lemma 4.2 in [4]. For the sake of notational convenience we denote 𝕃 = (dℙ/dℙ^{0}), L = 𝔼^{0}[𝕃/G] = (dP/dP^{0}), 𝕃^{0} = (dℙ^{0}/dℙ), and L^{0} = 𝔼[𝕃^{0}/G] = (dP^{0}/dP)

|𝔼^{0}[ψ/G] − 𝔼[ψ/G]| = |𝔼^{0}[ψ/G] − \frac{𝔼^{0}[𝕃ψ/G]}{𝔼^{0}[𝕃/G]}|

= \frac{1}{𝔼^{0}[𝕃/G]} |𝔼^{0}[ψ/G]𝔼^{0}[𝕃/G] − 𝔼^{0}[𝕃ψ/G]|

= \frac{1}{L} |𝔼^{0}[ψ/G]L − 𝔼^{0}[𝕃ψ/G]|

≤ \frac{1}{L} |𝔼^{0}[ψ/G]L − 𝔼^{0}[ψ/G]| + |𝔼^{0}[ψ/G] − 𝔼^{0}[𝕃ψ/G]|.

Moreover

|𝔼^{0}[ψ/G]L − 𝔼^{0}[ψ/G]| ≤ ∥ψ∥_{∞}|L − 1|,

and

|𝔼^{0}[ψ/G] − 𝔼^{0}[𝕃ψ/G]| ≤ 𝔼^{0}[ψ|1 − 𝕃|/G] ≤ ∥ψ∥_{∞}𝔼^{0}[|1 − 𝕃|/G].

Therefore

|𝔼^{0}[ψ/G] − 𝔼[ψ/G]| ≤ ∥ψ∥_{∞} \frac{1}{L} \{|L − 1| + 𝔼^{0}[|1 − 𝕃|/G]\}

( taking into account that 1/L = 1/(dP/dP^{0}) = (dP^{0}/dP) = L^{0})

= ∥ψ∥_{∞}L^{0} {|L − 1| + 𝔼^{0}[|1 − 𝕃|/G]} ,

and so

𝔼^{0} [|𝔼^{0}[ψ/G] − 𝔼[ψ/G]|] ≤ ∥ψ∥_{∞}𝔼^{0} [|L − 1| + 𝔼^{0}[|1 − 𝕃|/G]] .

Then the proof is achieved by noting that, by Jensen inequality,

𝔼^{0} [|L − 1|] = 𝔼^{0} [|(dP/dP^{0}) − 1|] = 𝔼^{0} [|𝔼^{0}[(dℙ/dℙ^{0}) − 1/G]|] ≤ 𝔼^{0} [|(dℙ^{0}/dℙ) − 1|] .

6 The M/M/1 queueing model: an approximation for the filter

As in Section 5, we suppose that under the measure P the process Q̃_{t}^{n} is an M/M/1 queue with parameters λ, λ (symmetric case), while, under the measure P^{n} the process Q̃_{t}^{n} is an M/M/1 queue with parameters λ_{n}, μ_{n} (general case).

The explicit form (22) for the filter of the discrete system is Σ̂^{n}(g)(s) (see (23)), evaluated in s = ξ_{t}^{n}, and it is evident from (25) that, from a computational point of view, it can be very hard to use it. Then the problem arises of finding a good approximation for this filter which, at the same time, is simpler to handle, and depends on the really observed trajectory, so that it can be actually used in applications.

A natural candidate is Π̂(g)(ξ_{t}^{n}), where Π̂(g)(s) is defined by (47). We will show in Theorem 6.1 that Π̂(g)(ξ_{t}^{n}), approximates the discrete filter in the L_{1} (Ω × [0, T])-norm, for each T > 0. The statement holds both for the symmetric case and for the general one.

Before stating formally the result we recall that Σ̂^{n}(g)(s) depends on the law of Q̃_{t}^{n}, or equivalently on the probability measure we use (see Remark 3.1). Then, to emphasize this dependence we write, in analogy with (23)

Σ̂_{P^{n}}^{n}(g)(s) = E^{n} [g(X_{s}^{n∗})/{L_{s}^{n∗} < b_{n}}] ,  (69)

and

Σ̂_{P}^{n} (g)(s) = E [g(X_{s}^{n∗})/{L_{s}^{n∗} < b_{n}}] .  (70)

Theorem 6.1. For all g bounded and continuous and for each T > 0

∫_{0}^{T} E|Σ̂_{P}^{n} (g)(ξ_{t}^{n}) − Π̂(g)(ξ_{t}^{n})|dt ⟶_{n→∞} 0.

∫_{0}^{T} E^{n}|Σ̂_{P^{n}}^{n}(g)(ξ_{t}^{n}) − Π̂(g)(ξ_{t}^{n})|dt ⟶_{n→∞} 0.

Proof. Consider first the symmetric case.

The limit we are looking for depends only on the distribution of ξ_{t}^{n}, therefore, as observed in Remark 5.4, using the Skorohod representation theorem as in the proof of Proposition 5.3, we can assume that

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

Proposition 5.5 implies that

Σ̂^{n}(g)(s_{n}) ≡ Σ̂_{P}^{n} (g)(s_{n}) ⟶_{n→∞} Π̂(g)(s),  (71)

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

Π̂(g)(s_{n}) ⟶_{n→∞} Π̂(g)(s),

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

Σ̂_{P}^{n} (g)(ξ_{t}^{n}) ⟶_{n→∞} Π̂(g)(ζ_{t})  and  Π̂(g)(ξ_{t}^{n}) ⟶_{n→∞} Π̂(g)(ζ_{t})  (72)

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 dP × dt, and an easy application of the dominated convergence theorem imply that

∫_{0}^{T} E [|Σ̂_{P}^{n} (g)(ξ_{t}^{n}) − Π̂(g)(ζ_{t})|^{p}] → 0, for any p > 0

and

∫_{0}^{T} E [|Σ̂_{P}^{n} (g)(ξ_{t}^{n}) − Π̂(g)(ξ_{t}^{n})|^{p}] → 0,  for any p > 0.  (73)

Consider now the asymmetric case.

Note that

∫_{0}^{T} E^{n} [|Σ̂_{P^{n}}^{n}(g)(ξ_{t}^{n}) − Π̂(g)(ξ_{t}^{n})|] dt
≤ ∫_{0}^{T} E^{n} [|Σ̂_{P^{n}}^{n}(g)(ξ_{t}^{n}) − Σ̂_{P}^{n} (g)(ξ_{t}^{n})|] dt + ∫_{0}^{T} E^{n} [|Σ̂_{P}^{n} (g)(ξ_{t}^{n}) − Π̂(g)(ξ_{t}^{n})|] dt

Therefore by Lemma 5.12

E^{n} [|Σ̂_{P^{n}}^{n}(g)(ξ_{t}^{n}) − Σ̂_{P}^{n} (g)(ξ_{t}^{n})|] = E^{n}[ |E^{n}[g(Q_{t}^{n})/G_{t}^{n}] − E[g(Q_{t}^{n})/G_{t}^{n}]| ]
≤ 2 ∥ g ∥_{∞} E^{n}[ |1 − (dP/dP^{n})|_{F_{t}^{n}} | ] = 2 ∥ g ∥_{∞} E[ |L_{t}^{n} − 1| ].

Moreover, by Cauchy-Schwartz inequality

E^{n} [|Σ̂_{P}^{n} (g)(ξ_{t}^{n}) − Π̂(g)(ξ_{t}^{n})|] = E [L_{t}^{n}|Σ̂_{P}^{n} (g)(ξ_{t}^{n}) − Π̂(g)(ξ_{t}^{n})|]
≤ E [(L_{t}^{n})^{2}] E [(Σ̂_{P}^{n} (g)(ξ_{t}^{n}) − Π̂(g)(ξ_{t}^{n}))^{2}] .

Summarizing

∫_{0}^{T} E^{n} [|Σ̂_{P^{n}}^{n}(g)(ξ_{t}^{n}) − Π̂(g)(ξ_{t}^{n})|] dt
≤2 ∥ g ∥_{∞} ∫_{0}^{T} E[ |L_{t}^{n} − 1| ]dt + ∫_{0}^{T} E [(L_{t}^{n})^{2}] E [(Σ̂_{P}^{n} (g)(ξ_{t}^{n}) − Π̂(g)(ξ_{t}^{n}))^{2}] dt,

and the thesis follows by the bounds (67) and (68) of Lemma 5.11 and by (73) of the symmetric case.

Remark 6.2. Note that

E^{n} [|Σ̂_{P^{n}}^{n}(g)(ξ_{t}^{n}) − Σ̂_{P}^{n} (g)(ξ_{t}^{n})|] = E^{n}[ |E^{n}[g(Q_{t}^{n})/G_{t}^{n}] − E[g(Q_{t}^{n})/G_{t}^{n}]| ]  (74)

is a particular case of the approximation problems considered in [4], where the observation process is a counting process: G_{t}^{n} coincides with the σ-algebra generated by the jump times of the observation process. Nevertheless in order to apply Theorem 2.1 in [4]) we need to assume the further and more restrictive assumptions that n(μ_{n} − λ) → 0 and n(λ − λ_{n}) → 0, which, under condition C3, may even converge to infinity.

7 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}  (75)

B̃_{t}^{n} = ∫_{0}^{t} 𝕀(Q̃_{s−}^{n} = 0)dÃ_{s}^{n}.  (76)

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}],  (77)

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 (77) in terms of γ_{t}^{0}(Q̃^{n}), is given in (84). 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}],  (78)

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 (78) under the same assumption C1, C2, C3 and A1, A2, A3 of Section 5. Under these assumptions we already know that Q_{t}^{n} converge weakly to a driftless Brownian motion W_{t} with diffusion coefficient 2λ. Moreover 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}. Consequently

(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 [12]) ^{2} . 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(g)(ζ_{t}). This result is proven in Theorem 7.4. Moreover Π̂(g)(γ_{t}^{0}(Q^{n})) is a good approximation of the filter for the rescaled model (see Theorem 7.6). Before giving the technical details we observe that the results of this section rely on the results of Sections 5 and 6. In these sections we have used the measures P in the symmetric case, and P^{n} otherwise. However in this section, as in Sections 2 and 3, we do not distinguish between

lim_{n→∞} E^{n} [μ_{n} ∫_{0}^{t} \sqrt{n}𝕀(Q_{s}^{n} = 0)ds] = E [Λ_{t}] .

2By the above considerations

C_{t}^{n} ⇒ Λ_{t} in D_{ℝ}([0, T])

and

C_{t}^{n} := \sqrt{n}μ_{n}C̃_{nt}^{n} = \sqrt{n}μ_{n} ∫_{0}^{t} 𝕀 (Q̃_{ns}^{n} = 0) ds = \sqrt{n}μ_{n} ∫_{0}^{t} 𝕀 (Q_{s}^{n} = 0) ds.

In general weak convergence does not imply immediately the convergence of the expected values, i.e. that Nevertheless, after some explicit calculations, it is possible to show that this is the case (see (95) in Appendix B).

the two cases for notational convenience, unless necessary.

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 7.1. 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. (see, for example,

[7] VI.7.3 page 187) the sequence {σ_{k+1}^{Bn}−σ_{k}^{Bn}}_{k≥0} is a sequence of mutually independent random variables and {σ_{k+1}^{Bn}−σ_{k}^{Bn}}_{k≥1} are also identically distributed. 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 7.1 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 7.2. 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)+  (79)

+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 (79) could be given by the techniques used in [5].

Proof. We begin by noting that 𝕀 (Q̃_{t}^{n} > 0) is H̃_{t}^{n}-adapted. Moreover we can state (see [3] 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 [13], 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 7.1. 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})}

(80)

= \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} ,  (81)

and that the process Q̃_{j,s}^{Bn} + 1 for s < sigma_{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) ]}.  (82)

As a consequence

E[g(Q̃_{t}^{n})/H̃_{t}^{n}] = 𝕀 (Q̃_{t}^{n} = 0) g(0)+  (83)
+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 (6),

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})} . (84)

The above considerations leads us to state the following result Theorem 7.3. 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) Σ̂^{n}(\overline{g}_{n})(γ_{t}^{0}(Q^{n})),  (85)

where Σ̂^{n} is the functional defined in (23), and \overline{g}_{n}(x) = g(x + \frac{1}{\sqrt{n}}). Proof. Equality (84) 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}(\overline{g}_{n})(s).

As a consequence of the above Theorem, for any g uniformly continuous

E[g(Q_{t}^{n})/H_{t}^{n}] = 𝕀 (Q_{t}^{n} = 0) g(0) + 𝕀 (Q_{t}^{n} > 0) Σ̂^{n}(\overline{g}_{n})(γ_{t}^{0}(Q^{n}))

(86)

= 𝕀 (Q_{t}^{n} = 0) g(0) + 𝕀 (Q_{t}^{n} > 0) Σ̂^{n}(g)(γ_{t}^{0}(Q^{n})) + ε(n, g) with

|ε(n, g)| ≤ ω_{g}(\frac{1}{\sqrt{n}}) = sup_{|x−y|≤ \frac{1}{\sqrt{n}}} |g(x) − g(y)|.  (87)

We can now state the main result of this section.

Theorem 7.4. For any g uniformly continuous

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

Proof. It is sufficient to prove that

𝕀 (Q_{t}^{n} = 0) ⟶^{P rob} 0,  for any t ≥ 0  (88)

and that

Σ̂^{n}(g)(γ_{t}^{0}(Q^{n})) ⇒ Π̂(g)(ζ_{t}),  for any t ≥ 0,  (89)

so that

𝕀 (Q_{t}^{n} = 0) g(0) + ε(n, g) ⟶^{P rob} 0

and

𝕀 (Q_{t}^{n} > 0) Σ̂^{n}(g)(γ_{t}^{0}(Q^{n})) ⇒ Π̂(g)(ζ_{t}),

since (88) is equivalent to 𝕀 (Q_{t}^{n} > 0) ⟶^{P rob} 1.

As recalled in (42), the sequence Q_{t}^{n} converges weakly to a reflected Brownian motion W_{t} + Λ_{t}. Then, the limit (88) 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.

To prove (89) we use the weak convergence of γ_{t}^{0}(Q^{n}) to ζ_{t} (see Proposition 5.3), and same techniques of the previous sections, in particular Remark 5.4, Proposition 5.5.

Remark 7.5. 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 6.1, namely for E[g(Q_{t}^{n})/H_{t}^{n}] in the symmetric case and for E^{n}[g(Q_{t}^{n})/H_{t}^{n}] in the asymptotically symmetric case. More precisely the following result holds

Theorem 7.6. For all g bounded and continuous and for each T > 0

∫_{0}^{T} E|E[g(Q_{t}^{n})/H_{t}^{n}] − Π̂(g)(γ_{t}^{0}(Q^{n}))|dt ⟶_{n→∞}0.

∫_{0}^{T} E^{n}|E^{n}[g(Q_{t}^{n})/H_{t}^{n}] − Π̂(g)(γ_{t}^{0}(Q^{n}))|dt ⟶_{n→∞} 0.

Proof. For the symmetric case, note that

|E[g(Q_{t}^{n})/H_{t}^{n}] − Π̂(g)(γ_{t}^{0}(Q^{n}))|^{p} =

= |𝕀 (Q_{t}^{n} = 0) g(0) + 𝕀 (Q_{t}^{n} > 0) Σ̂^{n}(g)(γ_{t}^{0}(Q^{n})) + ε(n, g) − Π̂(g)(γ_{t}^{0}(Q^{n}))|^{p}

≤ C(p)|𝕀 (Q_{t}^{n} = 0) (g(0) + Σ̂^{n}(g)(γ_{t}^{0}(Q^{n}))) + ε(n, g)|^{p} + C(p)|Σ̂^{n}(g)(γ_{t}^{0}(Q^{n})) − Π̂(g)(γ_{t}^{0}(Q^{n}))|^{p},

where C(p) is a suitable constant, and ε(n, g) is defined in (87). Then

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

The thesis follows since both the addends at the right hand side of the previous inequality converge to zero. The first addend at 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 6.1. The proof in the asymptotically symmetric case is analogous.

Acknowledgements We are indebted to Thomas G. Kurtz for the proof of Proposition 5.3.

A Appendix

A.1 Identification of the interpolating Brownian motion model

W_{t}^{n} defined by (32) admits the following representation

W_{t}^{n} = \frac{1}{2^{n}} V_{Z_{t}^{n}}^{n},  (90)

where

• Z_{t}^{n} is the point process defined by the sequence τ_{k}^{n};

• V_{k}^{n} is the discrete-time process satisfying the relation

V_{k}^{n} = V_{k−1}^{n} + U_{k}^{n},

where {U_{k}, k ∈ ℕ} is the sequence of {−1, 1}-valued random variables defined by the rule U_{k}^{n} = 2^{n} (W_{τ_{k}^{n}} − W_{τ_{k−1}^{n}}).

Let us consider the interarrival time sequence {T_{k}^{n} = τ_{k}^{n} − τ_{k−1}^{n}, k ≥ 1} of the point process Z_{t}^{n}. The following two lemmas are an easy consequence of the strong Markov property of W and of the fact that W and −W have the same law.

Lemma A.1. Z_{t}^{n} is a renewal process, that is {T_{k}^{n}, k ≥ 1} ia a sequence of i.i.d. random variables.

Lemma A.2. V_{k}^{n} is a symmetric random walk, i.e.

V_{k}^{n} = V_{k−1}^{n} + U_{k}^{n},

\{U_{k}^{n}, k ≥ 1\} is a sequence of i.i.d. random variables s.t. P (U_{k}^{n} = 1) = P (U_{k}^{n} = −1) = \frac{1}{2}

Moreover the following result can be found, for example, in [6] page 480 or [7] page 342.

Lemma A.3. The distribution function F^{n} of the random variables T_{k}^{n} is

F^{n}(t) = P (T_{k}^{n} ≤ t) = 4 ∑_{j=0}^{+∞}(−1)^{j} \frac{1}{\sqrt{2π}} ∫ +∞_{\frac{(2j+1)}{2^{n}\sqrt{t}}} exp (−\frac{x^{2}}{2} ) dx  (91)

and its generating function is

E [exp (−αT_{k}^{n})] = \frac{1}{cosh (\frac{\sqrt{2α}}{2^{n}})}.

Remark A.4. Note that the distribution function (91) can be written in the equivalent way

P (T_{k}^{n} ≤ t) = 4 ∑_{j=0}^{+∞}(−1)^{j}P (W_{t} ≥ \frac{(2j + 1)}{2^{n}}) .

Lemma A.5. The processes {Z_{t}^{n}, t ∈ ℝ^{+}} and {V_{k}^{n}, k ∈ ℕ} are mutually independent.

Proof. Note that, for each k ∈ ℕ, U_{k}^{n} and (T_{1}^{n}, ...T_{k−1}^{n}) are independent. In fact U_{k}^{n} and F_{τ_{k−1}^{n}} are independent and (T_{1}^{n}, ...T_{k−1}^{n}) are F_{τ_{k−1}^{n}}-measurable. Moreover U_{k}^{n} is F_{τ_{k}^{n}}-measurable, so it is independent of (T_{k+1}^{n}, T_{k+2}^{n}, ...).

So we just have to prove the independence between U_{k}^{n} and T_{k}^{n}. We consider the case k = 1 taking into account that this procedure easily extends to the general case.

To this end it will suffices to verify

E [𝕀 (U_{1} = 1) exp(−ατ_{1}^{n})] = E [𝕀 (U_{1} = 1)] E [exp(−ατ_{1}^{n})] .  (92)

Setting

y_{−1}(α) =E [𝕀 (U_{1} = −1) exp(−ατ_{1}^{n})]
y_{1}(α) =E [𝕀 (U_{1} = 1) exp(−ατ_{1}^{n})]

by a direct computation we get

y_{1}(α) = y_{−1}(α) = \frac{1}{2 cosh (\frac{\sqrt{2α}}{2^{n}})}

and then

y_{1}(α) = y_{1}(0)[y_{1}(α) + y_{−1}(α)]  (93)

which is equivalent to (92).

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

B A limit result and an upper bound for the rescaled queue

In this section we prove that, under conditions C1, C2, and C3

E[Λ_{t}] = lim_{n→∞} E^{n} [μ_{n} ∫_{0}^{t} \sqrt{n}𝕀(Q_{s^{−}}^{n} = 0)ds] = lim_{n→∞} μ_{n} ∫_{0}^{t} \sqrt{n}P^{n}(Q_{s}^{n} = 0/Q_{0}^{n} = 0)ds, (94)

where Λ_{t} is the local time of a Wiener process with diffusion coefficient 2λ and drift coefficient 0, and moreover we prove that for any 1 < α < 2, there exists n_{α} such that for any n ≥ n_{α}

∫_{0}^{t} \sqrt{n}P^{n}(Q_{s}^{n} = 0/Q_{0}^{n} = 0)ds ≤ n^{(1−α)/2} + \sqrt{n}(1 − \frac{λ_{n}}{μ_{n}} ) t + K\sqrt{t},  (95)

for a computable constant K. As a consequence, since \sqrt{n} (1 − \frac{λ_{n}}{μ_{n}} ) = \frac{1}{μ_{n}} \sqrt{n} (μ_{n} − λ_{n}) → 0, under assumption C3,

lim sup_{n→∞} ∫_{0}^{t} \sqrt{n}P^{n}(Q_{s}^{n} = 0/Q_{0}^{n} = 0)ds ≤ K\sqrt{t}.

To this end we start by recalling a few properties that we need, then we sketch the proof of the upper bound and of the limit, and finally we present the technical details.

B.1 Preliminary results

1) Modified Bessel functions The modified Bessel functions of order k, for k ≥ 0 are defined as

I_{k}(x) = ∑_{i=0}^{∞} \frac{1}{(k + i)!i!} (\frac{x}{2})^{k+2i} ,

while, for for k ≤ 0

I_{k}(x) = I_{−k}(x).

2) The asymptotic behaviour of modified Bessel functions It is well known that, for any k,

I_{k}(x) ∼ \frac{e^{x}}{\sqrt{2π x}}  when x → ∞,  (96)

and therefore, for any k, there exist an \overline{x}_{k} > 0 and an L_{k} such that

sup_{x≥\overline{x}_{k}} \frac{I_{k}(x)}{\frac{e^{x}}{\sqrt{2π x}}} ≤ L_{k}  (97)

3) The one-dimensional distribution of continuous time random walk Let Z_{t} be a standard Poisson process and Y_{t} = V_{Z_{t}}, where as usual {V_{k}, k ∈ ℕ} is a discrete time random walk, such that V_{0} = 0, and V_{k} = V_{k−1} + U_{k}, where {U_{k}, k ∈ ℕ} is a sequence of independent {−1, 1}-valued random variables with P (U_{k} = 1) = p and P (U_{k} = −1) = q. Then, for any u > 0 (see, for instance, [7] II.7 page 58)

P (Y_{u} = k) = exp\{−u\} (\frac{p}{q})^{\frac{k}{2}} I_{k}(2\sqrt{pq}u).

As a consequence, for any u > 0 and any positive numbers p, q ∈ (0, 1), with p + q = 1

∑_{k=−∞}^{∞} exp\{−u\} (\frac{p}{q})^{\frac{k}{2}} I_{k}(2\sqrt{pq}u) = 1,

and therefore

∑_{k=2}^{∞} exp\{−u\} (\frac{p}{q})^{\frac{k}{2}} I_{k}(2\sqrt{pq}u) ≤ 1,

and, taking into account that I_{k}(x) = I_{−k}(x) (or equivalently interchanging the role of p and q)

∑_{k=2}^{∞} exp\{−u\} (\frac{q}{p})^{\frac{k}{2}} I_{k}(2\sqrt{pq}u) ≤ 1,  (98)

4) Transition probabilities p_{k,m}(t) of a birth and death process with reflection in l Assume that Q_{t} is a birth and death process, assume that the upward jump rate is ν and the downward jump rate is ρ, and that the process lives in [l, ∞) ∩ ℕ, with l a reflecting state. Then the transition probabilities (see e.g. [8]) are

p_{k,m}(t) = P (Q_{t} = m/Q_{0} = k) = (\frac{ν}{ρ})^{\frac{k−m}{2}} exp\{−t(ν + ρ)\} [I_{k−m}(2\sqrt{ρν} t)
+ (\frac{ρ}{ν})^{\frac{1}{2}} I_{m+k+1−2l}(2\sqrt{ρν} t)

+ (1 − \frac{ν}{ρ}) ∑_{j=2}^{∞} (\frac{ρ}{ν})^{\frac{j}{2}} I_{m+k−2l+j}(2\sqrt{ρν} t)]

5) The behaviour of p_{0,0}(t) with reflection in 0 When k = m = l = 0, then

p_{0,0}(t) = P (Q_{t} = 0/Q_{0} = 0) = exp\{−t(ν + ρ)\}[I_{0}(2\sqrt{ρν} t) + (\frac{ρ}{ν})^{\frac{1}{2}} I_{1}(2\sqrt{ρν} t)

+ (1 − \frac{ν}{ρ}) ∑_{j=2}^{∞} (\frac{ρ}{ν})^{\frac{j}{2}} I_{j}(2\sqrt{ρν} t).]

p_{0,0}(t) − exp\{−t(ν + ρ)\}[I_{0}(2\sqrt{ρν} t) + (\frac{ρ}{ν})^{\frac{1}{2}} I_{1}(2\sqrt{ρν} t)]

= (1 − \frac{ν}{ρ})[ exp\{−t(ν + ρ)\} ∑_{j=2}^{∞} (\frac{ρ}{ν})^{\frac{j}{2}} I_{j}(2\sqrt{ρν} t).]

Note that if we set t = ν +ρ , p = \frac{ν}{ν+ρ}, and q = \frac{ρ}{ν+ρ}, then the expression in square brackets in the previous expression is identical to (98) with u = (ν +ρ)t, and therefore it is bounded by 1. As a consequence we have that

|p_{0,0}(t) − exp\{−t(ν + ρ)\}[I_{0}(2\sqrt{ρν} t) + (\frac{ρ}{ν})^{\frac{1}{2}} I_{1}(2\sqrt{ρν} t)]| ≤ (1 − \frac{ν}{ρ})  (99)

and in particular

p_{0,0}(t) ≤ exp\{−t(ν + ρ)\}[I_{0}(2\sqrt{ρν} t) + (\frac{ρ}{ν})^{\frac{1}{2}} I_{1}(2\sqrt{ρν} t)] + (1 − \frac{ν}{ρ}).  (100)

B.2 Sketch of the proof

In our case we are interested to give an upper bound for

E^{n} [∫_{0}^{t} \sqrt{n}𝕀(Q_{s}^{n} = 0)ds] = ∫_{0}^{t} \sqrt{n}P^{n}(Q_{s}^{n} = 0/Q_{0}^{n} = 0)ds.

We can use (100) in the case when ν = nλ_{n}, and ρ = nμ_{n} and get

\sqrt{n}P^{n}(Q_{s}^{n} = 0/Q_{0}^{n} = 0)

≤ \sqrt{n} exp\{−sn(λ_{n} + μ_{n})\}[I_{0}(x_{n}(s)) + (\frac{μ_{n}}{λ_{n}} )^{\frac{1}{2}} I_{1}(x_{n}(s))] + \sqrt{n}(1 − \frac{λ_{n}}{μ_{n}} ),

where

x_{n}(s) = 2n\sqrt{μ_{n}λ_{n}} s.

For any real number α > 1

∫_{0}^{t} \sqrt{n}P^{n}(Q_{s}^{n} = 0/Q_{0}^{n} = 0)ds = ∫_{0}^{n^{−α/2}} \sqrt{n}P^{n}(Q_{s}^{n} = 0/Q_{0}^{n} = 0)ds + ∫_{n^{−α/2}}^{t} \sqrt{n}P^{n}(Q_{s}^{n} = 0/Q_{0}^{n} = 0)ds

≤ n^{(1−α)/2} + ∫_{n^{−α/2}}^{t} \sqrt{n} exp\{−sn(λ_{n} + μ_{n})\}[I_{0}(x_{n}(s)) + (\frac{μ_{n}}{λ_{n}} )^{\frac{1}{2}} I_{1}(x_{n}(s))]ds

+ \sqrt{n}(1 − \frac{λ_{n}}{μ_{n}} ) t.

Then, using (97), it is possible to show (see the technical details at the end of this section) that

for any 1 < α < 2 and for any n^{1−α/2} ≥ \frac{max(\overline{x}_{0}, \overline{x}_{1})}{2\sqrt{μ_{n}λ_{n}}}

the following bound holds

∫_{0}^{t} \sqrt{n}P^{n}(Q_{s}^{n} = 0/Q_{0}^{n} = 0)ds ≤ n^{(1−α)/2} + \sqrt{n}(1 − \frac{λ_{n}}{μ_{n}} ) t

+ ∫_{n^{−α/2}}^{t} \frac{1}{2\sqrt{π\sqrt{μ_{n}λ_{n}}s}} [L_{0} + (\frac{μ_{n}}{λ_{n}} )^{\frac{1}{2}} L_{1}] ds.

Starting from the bound (99) instead of (100), and using (96), the same technique can be used to show that

lim_{n→∞} E^{n} [∫_{0}^{t} \sqrt{n}μ_{n}𝕀(Q_{s^{−}}^{n} = 0)ds] = lim_{n→∞} μ_{n} ∫_{n^{−α/2}}^{t} \frac{1}{2\sqrt{π\sqrt{μ_{n}λ_{n}}s}} [1 + (\frac{μ_{n}}{λ_{n}} )^{\frac{1}{2}} ] ds = E[Λ_{t}]

where Λ_{t} is the local time of a Wiener process with diffusion coefficient 2λ and drift coefficient 0. Indeed, on the one hand

lim_{n→∞} μ_{n} ∫_{n^{−α/2}}^{t} \frac{1}{2\sqrt{π\sqrt{μ_{n}λ_{n}}s}} [1 + (\frac{μ_{n}}{λ_{n}} )^{\frac{1}{2}} ] ds = λ ∫_{0}^{t} \frac{1}{2\sqrt{λπs}} [1 + (\frac{λ}{λ})^{\frac{1}{2}} ] ds

= ∫_{0}^{t} \begin{matrix}√ & \overline{λ} \\ √ & \overline{πs} \end{matrix}ds = \frac{2\sqrt{λt}}{\sqrt{π}},

On the other hand, by the reflection principle, the density of the random variable Λ_{t} is

f_{Λ_{t}}(u) = \frac{e^{− \frac{u^{2}}{4λt}}}{\sqrt{πλt}}𝕀(u > 0),

and then

E[Λ_{t}] = \frac{1}{\sqrt{πλt}} ∫_{0}^{∞} ue^{− \frac{u^{2}}{4λt}} du = \frac{2\sqrt{λt}}{\sqrt{π}}.

B.3 Technical details

Taking into account (97), we get, for s ∈ [n^{−α/2}, t] \sqrt{n}P^{n}(Q_{s}^{n} = 0/Q_{0}^{n} = 0)

≤ \sqrt{n} exp\{−sn(λ_{n} + μ_{n})\}\frac{exp\{x_{n}(s)\}}{\sqrt{2πx_{n}(s)}} [ sup_{u∈[n^{−α/2},t]} \frac{I_{0}(x_{n}(u))}{\frac{exp\{x_{n}(u)\}}{\sqrt{2πx_{n}(u)}}} + (\frac{μ_{n}}{λ_{n}} )^{\frac{1}{2}} sup_{u∈[n^{−α/2},t]} \frac{I_{1}(x_{n}(u))}{\frac{exp\{x_{n}(u)\}}{\sqrt{2πx_{n}(u)}}} ]

Now

\sqrt{n} exp\{−sn(λ_{n} + μ_{n})\}\frac{exp\{x_{n}(s)\}}{\sqrt{2πx_{n}(s)}} = \sqrt{n} exp\{−sn[(λ_{n} + μ_{n}) − 2\sqrt{μ_{n}λ_{n}} ]\}\frac{1}{\sqrt{4πn\sqrt{μ_{n}λ_{n}}s}}

= exp\{−sn[(λ_{n} + μ_{n}) − 2\sqrt{μ_{n}λ_{n}} ]\}\frac{1}{2\sqrt{π\sqrt{μ_{n}λ_{n}}s}} ≤ \frac{1}{2\sqrt{π\sqrt{μ_{n}λ_{n}}s}}

since (λ_{n} + μ_{n}) − 2\sqrt{μ_{n}λ_{n}} ≥ 0.

Moreover, for any n, and for k = 0, 1

sup_{u∈[n^{−α/2},t]} \frac{I_{k}(x_{n}(u))}{\frac{exp\{x_{n}(u)\}}{\sqrt{2πx_{n}(u)}}} = sup_{x∈[x_{n}(n^{−α/2},x_{n}(t)]} \frac{I_{k}(x)}{\frac{exp\{x\}}{\sqrt{2πx}}} ≤ sup_{x≥2n\sqrt{μ_{n}λ_{n}}n^{−α/2}} \frac{I_{0}(x)}{\frac{exp\{x\}}{\sqrt{2πx}}}

and therefore, with the same notations of (97), for α < 2, for n sufficiently large so that 2n^{1−α/2}\sqrt{μ_{n}λ_{n}} ≥ \overline{x}_{k}, we get

sup_{u∈[n^{−α/2},t]} \frac{I_{k}(x_{n}(u))}{\frac{exp\{x_{n}(u)\}}{\sqrt{2πx_{n}(u)}}} ≤ L_{k}

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