# Moderate deviations of many–server queues
in the Halfin–Whitt regime
and weak convergence methods

Anatolii A. Puhalskii

##  1 Introduction

Many–server queues are important in applications but their analysis beyond
Markovian assumptions is difficult, see, e.g., Asmussen [2]. As a way to gain
insight into the operation of many–server queues, various heavy–traffic
asymptotics have been explored when the arrival and service rates tend to
infinity. Iglehart and Whitt [14] considered the number of servers, $n$ ,
fixed, assumed that the traffic intensity, $\rho$ , tends to unity and showed
that properly normalized version of the queue length process converges to a
one dimensional reflected Brownian motion, in analogy with the heavy–traffic
limit for a $GI/GI/1$ queue. As for the single server queue, the probability
of having at least one customer in the queue converges to unity. Many server
queues with the numbers of servers going to infinity for different initial
conditions and assumptions on the interarrival and service time distributions,
were considered by Borovkov [5, 6], Iglehart [12, 13] and Whitt [31]. The
growth conditions imposed on $n$ result in the queue being asymptotically
equivalent to an infinite server queue so that the probability of wait tends
to zero.

In order to capture the situation of the probability of a customer wait being
strictly between zero and unity, Halfin and Whitt [11] proposed the setup
where service time distributions are held fixed, while the number of servers
and the arrival rate grow without a bound in such a way that
$\sqrt{n}(1-\rho)\to\beta\in\mathbb{R}\,.$ With $Q(t)$ denoting the number of
customers present at time $t$ , assuming the initial conditions are suitably
chosen, the sequence of processes $(Q(t)-n)/\sqrt{n}$ , considered as random
elements of the associated Skorohod space, converges in law to a
continuous–path process. In their pioneering contribution, Halfin and Whitt
[11] considered the case of exponentially distributed service times. The limit
process then is an Ito diffusion with a piecewise linear drift. An extension
to generally distributed service times is more recent and is due to Reed [29],
see also Puhalskii and Reed [28]. The limit process, however, is no longer
generally Markov and is not well understood, cf. Kaspi and Ramanan [17] and
Aghajani and Ramanan [1], for related results.

In search of more tractable asymptotics, it was proposed in Puhalskii [27] to
study moderate deviations of $Q(t)$ and a Large Deviation Principle (LDP) was
conjectured for the process $(Q(t)-n)/(b_{n}\sqrt{n})$ under the heavy traffic
condition $\sqrt{n}/b_{n}(1-\rho)\to\beta$ , where $b_{n}\to\infty$ and
$b_{n}/n\to 0$ . The deviation function (a.k.a. rate function) was purported
to solve a convex variational problem with a quadratic objective function.
This paper confirms the conjecture and proves the LDP in question.
Furthermore, the deviation function is expressed in terms of the solution to a
Fredholm equation of the second kind which could facilitate evaluating the
deviation function numerically. It is noteworthy that although the LDP
generally produces cruder asymptotics than weak convergence, or convergence in
law, it may capture exponents in the distributions of corresponding weak
convergence limits, cf. Puhalskii [25].

The main thrust of the proofs is to demonstrate the utility of weak
convergence methods for deriving large deviation asymptotics. As in Puhalskii
[24, 26], the LDP is viewed as an analogue of weak convergence, the
cornerstone of the approach being the following analogue of celebrated
Prohorov’s tightness theorem: “A sequence of probability measures on a
complete separable metric space is exponentially tight if and only if every
its subsequence admits a further subsequence that satisfies the LDP” (Theorem
(P) in Puhalskii [23]). Consequently, once exponential tightness has been
proved, the proof of the LDP is achieved by identifying a large deviation
limit point. A possible identification tool is an analogue of the method of
finite–dimensional distributions. It is what is used in this paper in order to
derive logarithmic asymptotics of moderate deviations of the sequential
empirical process, which is a key element of the proof of the main result, on
the one hand, and complements developments in Krichagina and Puhalskii [18]
and in Puhalskii and Reed [28], on the other hand. In order to use the analogy
with weak convergence to the utmost, this paper formulates the definition of
the LDP for stochastic processes as large deviation (LD) convergence to
idempotent processes, see Puhalskii [26] and Appendix A for definitions and
properties. With tools for the study of weak convergence properties of
many–server queues in heavy traffic being well developed, the moderate
deviation asymptotics are derived by using similar methods, mutatis mutandis,
that proved their worth in the weak convergence context. An important
distinction on the technical side is an extensive use of exponential
martingales, as establishing exponential tightness is a more challenging task
than establishing tightness. In addition, a study of idempotent counterparts
of the standard Wiener process, the Brownian bridge and the Kiefer process is
carried out. The properties of those idempotent processes are integral to the
proofs.

The paper’s organisation is as follows. In Section 2, a precise specification
of the model is provided, as well as the main result on the logarithmic
asymptotics of moderate deviations of the number–in–the–system process. The
latter is stated both in the form of a trajectorial LDP and in the form of LD
convergence to an idempotent process. The proofs are given in Section 3.
Section 4 is concerned with evaluating the deviation function by reduction to
solving a Fredholm equation of the second kind. For the reader’s convenience,
Appendix A gives a primer on idempotent processes and the use of weak
convergence methods for proving the LDP. Appendix B is concerned with absolute
continuity of the solution to a nonlinear renewal equation which is needed in
Section 4.

##  2 Trajectorial moderate deviation limit theorems

Assume as given a sequence of many server queues with unlimited waiting room
indexed by $n$, where $n$ represents the number of servers. Service is
performed on a first–come–first–served basis. Let $Q_{n}(t)$ denote the number
of customers present at time $t$ . Out of those, $Q_{n}(t)\wedge n$ customers
are in service and $(Q_{n}(t)-n)^{+}$ customers are in the queue. The service
times of the customers in the queue at time $0$ and the service times of
customers exogenously arriving after time $0$ are denoted by
$\eta_{1},\eta_{2},\ldots$ in the order of entering service and come from an
i.i.d. sequence of positive unbounded random variables with continuous
distribution function $F$ . In terms of $F$ , it is thus assumed that

$F(0)=0\,,F(x)<1\,,\text{ for all }x\,.$ (2.1)

The mean service time $\mu^{-1}=\int_{0}^{\infty}x\,dF(x)$ is assumed to be
finite. Residual service times of the customers in service at time $0$ are
denoted by $\eta^{(0)}_{1},\eta^{(0)}_{2},\ldots$ and are assumed to be i.i.d.
with distribution $F_{0}$ that is the distribution of the delay in a
stationary renewal process with inter–renewal distribution $F$ . Thus,

$F_{0}(x)=\mu\int_{0}^{x}(1-F(y))\,dy\,.$ (2.2)

Let $A_{n}(t)$ denote the number of exogenous arrivals by $t$ , with
$A_{n}(0)=0$ . It is assumed that $A_{n}(t)$ has unit jumps. The entities
$Q_{n}(0)$, $\\{\eta^{(0)}_{1},\eta^{(0)}_{2},\ldots\\}$,
$\\{\eta_{1},\eta_{2},\ldots\\}$, and $A_{n}=(A_{n}(t),\,t\in\mathbb{R}_{+})$
are assumed to be independent. All stochastic processes are assumed to have
rightcontinuous paths with lefthand limits. Let ${{\hat{A}}_{n}}(t)$ denote
the number of customers that enter service after time $0$ and by time $t$ ,
with ${{\hat{A}}_{n}}(0)=0$ . Since the distribution function of $\eta_{i}$ is
continuous, $\hat{A}_{n}$ has unit jumps a.s. By considering the jumps of the
processes on both sides of the next equation, one can see that

$(Q_{n}(t)-n)^{+}+{\hat{A}}_{n}(t)=(Q_{n}(0)-n)^{+}+A_{n}(t)\,.$ (2.3)

In addition, balancing the arrivals and departures yields the equation

$Q_{n}(t)=Q_{n}^{(0)}(t)+(Q_{n}(0)-n)^{+}+A_{n}(t)-\int_{0}^{t}\int_{0}^{t}\mathbf{1}_{\\{s+x\leq
t\\}}\,d\,\sum_{i=1}^{{{\hat{A}}_{n}}(s)}\mathbf{1}_{\\{\eta_{i}\leq x\\}}\,,$
(2.4)

where

$Q_{n}^{(0)}(t)=\sum_{i=1}^{Q_{n}(0)\wedge
n}\,\,\mathbf{1}_{\\{\eta^{(0)}_{i}>t\\}}\,\,,$ (2.5)

which quantity represents the number of customers present at time $t$ out of
those in service at time $0$ , and

$\int_{0}^{t}\int_{0}^{t}\mathbf{1}_{\\{s+x\leq
t\\}}\,d\,\sum_{i=1}^{{{\hat{A}}_{n}}(s)}\mathbf{1}_{\\{\eta_{i}\leq
x\\}}=\sum_{i=1}^{\hat{A}_{n}(t)}\,\mathbf{1}_{\\{\hat{\tau}_{n,i}+\eta_{i}\leq
t\\}}\,$ (2.6)

which quantity represents the number of customers that enter service after
time $0$ and depart by time $t$ , $\hat{\tau}_{n,i}$ denoting the $i$th jump
time of $\hat{A}_{n}$ , i.e.,

$\hat{\tau}_{n,i}=\inf\\{t:\,\hat{A}_{n}(t)\geq i\\}\,.$ (2.7)

For existence and uniqueness of solutions to (2.3)–(2.5), the reader is
referred to Puhalskii and Reed [28].

Given a sequence $r_{n}\to\infty$ , as $n\to\infty$ , a sequence $P_{n}$ of
probability laws on the Borel $\sigma$–algebra of a metric space $M$ and a
$[0,\infty]$–valued function $I$ on $M$ such that the sets $\\{y\in
M:\,I(y)\leq\gamma\\}$ are compact for all $\gamma\geq 0$ , the sequence
$P_{n}$ is said to obey the LDP for rate $r_{n}$ with deviation function $I$
provided $\lim_{n\to\infty}1/r_{n}\,\ln P_{n}(W)=-\inf_{y\in W}I(y)\,,$ for
every Borel set $W$ such that the infima of $I$ over the interior and closure
of $W$ agree.

The deviation function for the queue length process is introduced next. For
$T>0$ and $m\in\mathbb{N}$ , let $\mathbb{D}([0,T],\mathbb{R}^{m})$ and
$\mathbb{D}(\mathbb{R}_{+},\mathbb{R}^{m})$ represent the respective Skorohod
spaces of rightcontinuous $\mathbb{R}^{m}$–valued functions with lefthand
limits defined on $[0,T]$ and $\mathbb{R}_{+}$ , respectively. These spaces
are endowed with metrics rendering them complete separable metric spaces, see
Ethier and Kurtz [9], Jacod and Shiryaev [15], for more detail. Given
$q=(q(t)\,,t\in\mathbb{R}_{+})\in\mathbb{D}(\mathbb{R}_{+},\mathbb{R})$ , let

$I^{Q}(q)=\frac{1}{2}\inf\\{\int_{0}^{1}\dot{w}^{0}(x)^{2}\,dx+\int_{0}^{\infty}\dot{w}(t)^{2}\,dt+\int_{0}^{\infty}\int_{0}^{1}\dot{k}(x,t)^{2}\,dx\,dt\\}\,,$
(2.8)

the $\inf$ being taken over
$w^{0}=(w^{0}(x)\,,x\in[0,1])\in\mathbb{D}([0,1],\mathbb{R}),w=(w(t)\,,t\in\mathbb{R}_{+})\in\mathbb{D}(\mathbb{R}_{+},\mathbb{R})$
and
$k=((k(x,t)\,,x\in[0,1])\,,t\in\mathbb{R}_{+})\in\mathbb{D}(\mathbb{R}_{+},\mathbb{D}([0,1],\mathbb{R}_{+}))$
such that $w^{0}(0)=w^{0}(1)=0$ , $w(0)=0$ , $k(x,0)=k(0,t)=k(1,t)=0$ ,
$w^{0}$, $w$ and $k$ are absolutely continuous with respect to Lebesgue
measures on $[0,1]$ , $\mathbb{R}_{+}$ and $[0,1]\times\mathbb{R}_{+}$ ,
respectively, and, for all $t$ , with $q_{0}^{-}=(-q_{0})^{+}$ ,

$q(t)=(1-F(t))q_{0}^{+}-(1-F_{0}(t))q_{0}^{-}-\beta
F_{0}(t)+\int_{0}^{t}q(t-s)^{+}\,dF(s)+w^{0}(F_{0}(t))\\\
+\int_{0}^{t}\bigl{(}1-F(t-s)\bigr{)}\sigma\,\dot{w}(s)\,ds+\int_{\mathbb{R}_{+}^{2}}\,\mathbf{1}_{\\{s+x\leq
t\\}}\,\,\dot{k}(F(x),\mu s)\,dF(x)\,\mu\,ds\,,$ (2.9)

where $\dot{w}^{0}(x)$ , $\dot{w}(t)$ and $\dot{k}(x,t)$ represent the
respective Radon–Nikodym derivatives. If $w^{0}$ , $w$ and $k$ as indicated do
not exist, then $I^{Q}(q)=\infty$ . Consequently, $I^{Q}(q)=\infty$ unless
$q(t)$ is continuous as the righthand side of (2.9) is a continuous function
of $t$ . (It is proved in Lemma B.1 that if $F$ is, in addition, absolutely
continuous with respect to Lebesgue measure, then $q(t)$ in (2.9) is
absolutely continuous too.) By Lemma B.1 in Puhalskii and Reed [28], the
equation in (2.9) has a unique solution for $q(t)$ as an element of
$\mathbb{L}_{\infty,\text{loc}}(\mathbb{R}_{+},\mathbb{R})$ .

Let the processes $X_{n}=(X_{n}(t),t\in\mathbb{R}_{+})$ be defined by

$X_{n}(t)=\frac{\sqrt{n}}{b_{n}}\bigl{(}\frac{Q_{n}(t)}{n}-1\bigr{)}\,.$
(2.10)

The next theorem confirms a conjecture in Puhalskii [27].

######  Theorem 2.1.

Suppose, in addition, that $A_{n}$ is a renewal process of rate $\lambda_{n}$
. Let $\rho_{n}=\lambda_{n}/(n\mu)$ , $\beta\in\mathbb{R}$ ,
$q_{0}\in\mathbb{R}$ and $\sigma>0$ . Suppose that, as $n\to\infty$,

$\frac{\sqrt{n}}{b_{n}}(1-\rho_{n})\to\beta$ (2.11)

and the sequence of random variables $\sqrt{n}/b_{n}\,(Q_{n}(0)/n-1)$ obeys
the LDP in $\mathbb{R}$ for rate $b_{n}^{2}$ with deviation function
$I_{q_{0}}(y)$ such that $I_{q_{0}}(q_{0})=0$ and $I_{q_{0}}(y)=\infty$ , for
$y\not=q_{0}$ . Suppose that the sequence of processes
$\bigl{(}\sqrt{n}/b_{n}(A_{n}(t)/n-\mu\rho_{n}t),t\in\mathbb{R}_{+}\bigr{)}$
obeys the LDP in $\mathbb{D}(\mathbb{R}_{+},\mathbb{R})$ for rate $b_{n}^{2}$
with deviation function $I^{A}(a)$ such that
$I^{A}(a)=1/(2\sigma^{2})\int_{0}^{\infty}\dot{a}(t)^{2}\,dt$ , provided
$a=(a(t)\,,t\in\mathbb{R}_{+})$ is an absolutely continuous nondecreasing
function with $a(0)=0$ , and $I^{A}(a)=\infty$ , otherwise. If, in addition,

$b_{n}^{3}n^{1/b_{n}^{2}-1/2}\to 0\,,$ (2.12)

then the sequence $X_{n}$ obeys the LDP in
$\mathbb{D}(\mathbb{R}_{+},\mathbb{R})$ for rate $b_{n}^{2}$ with deviation
function $I^{Q}(q)$ .

######  Remark 2.1.

In order that the LDP for
$\bigl{(}\sqrt{n}/b_{n}(A_{n}(t)/n-\mu\rho_{n}t),t\geq 0\bigr{)}$ in the
statement hold, it suffices that $E(n\xi_{n})\to 1/\mu$ ,
$\text{Var}(n\xi_{n})\to\sigma^{2}/\mu^{3}$ and that either
$\sup_{n}E(n\xi_{n})^{2+\epsilon}<\infty$ , for some $\epsilon>0$ , and
$\sqrt{\ln n}/b_{n}\to\infty$ or
$\sup_{n}E\exp(\alpha(n\xi_{n})^{\delta})<\infty$ and
$n^{\delta/2}/b_{n}^{2-\delta}\to\infty$ , for some $\alpha>0$ and
$\delta\in(0,1]$ , where $\xi_{n}$ represents a generic interarrival time for
the $n$-th queue, see Puhalskii [25].

LD convergence versions come next. Introduce

$Y_{n}(t)=\frac{\sqrt{n}}{b_{n}}\,\bigl{(}\frac{A_{n}(t)}{n}-\mu t\bigr{)}$
(2.13)

and let $Y_{n}=(Y_{n}(t),\,t\in\mathbb{R}_{+})$ . For the statement of the
next theorem, Appendix A is a recommended reading.

######  Theorem 2.2.

  1. 1.

Suppose that, as $n\to\infty$, the sequence $X_{n}(0)$ LD converges in
distribution in $\mathbb{R}$ at rate $b_{n}^{2}$ to an idempotent variable
$X(0)$ , the sequence $Y_{n}$ LD converges in distribution in
$\mathbb{D}(\mathbb{R}_{+},\mathbb{R})$ at rate $b_{n}^{2}$ to an idempotent
process $Y$ with continuous paths and (2.12) holds. Then the sequence $X_{n}$
LD converges in distribution in $\mathbb{D}(\mathbb{R}_{+},\mathbb{R})$ at
rate $b_{n}^{2}$ to the idempotent process $X=(X(t),t\in\mathbb{R}_{+})$ that
is the unique solution to the equation

$X(t)=(1-F(t))X(0)^{+}-(1-F_{0}(t))X(0)^{-}+\int_{0}^{t}X(t-s)^{+}\,dF(s)+W^{0}(F_{0}(t))\\\
+Y(t)-\int_{0}^{t}Y(t-s)\,dF(s)+\int_{0}^{t}\int_{0}^{t}\,\mathbf{1}_{\\{s+x\leq
t\\}}\,\,\dot{K}(F(x),\mu s)\,dF(x)\,\mu\,ds\,,$ (2.14)

where $W^{0}=(W^{0}(x),x\in[0,1])$ is a Brownian bridge idempotent process and
$K=(K(x,t),(x,t)\in\mathbb{R}_{+}\times[0,1])$ is a Kiefer idempotent process,
$X(0)$, $Y$ , $W^{0}$ and $K$ being independent.

  2. 2.

In the special case where $A_{n}$ is a renewal process of rate $\lambda_{n}$ ,
condition (2.11) holds and the sequence
$(\sqrt{n}/b_{n}\,(A_{n}(t)-\mu\rho_{n}t)\,,t\in\mathbb{R}_{+})$ LD converges
in distribution in $\mathbb{D}(\mathbb{R}_{+},\mathbb{R})$ at rate $b_{n}^{2}$
to $\sigma W$ , with $W=(W(t)\,,t\in\mathbb{R}_{+})$ being a standard Wiener
idempotent process, the limit idempotent process $X$ solves the equation

$X(t)=(1-F(t))X(0)^{+}-(1-F_{0}(t))X(0)^{-}-\beta
F_{0}(t)+\int_{0}^{t}X(t-s)^{+}\,dF(s)\\\
+W^{0}(F_{0}(t))+\int_{0}^{t}\bigl{(}1-F(t-s)\bigr{)}\sigma\,\dot{W}(s)\,ds+\int_{0}^{t}\int_{0}^{t}\,\mathbf{1}_{\\{s+x\leq
t\\}}\,\,\dot{K}(F(x),\mu s)\,dF(x)\,\mu\,ds\,,$ (2.15)

where $X(0)$, $W$, $W^{0}$ and $K$ are independent.

It is noteworthy that the LD convergence of the sequence $X_{n}$ in part 2 is
equivalent to the assertion of Theorem 2.1 . Besides, the limit idempotent
process in (2.14) is analogous to the limit stochastic process in Puhalskii
and Reed [28, p.139].

##  3 Proofs of Theorems 2.1 and 2.2

This section is mostly concerned with the proof of part 1 of Theorem 2.2. Part
2 and Theorem 2.1 are easy consequences. A proof outline is as follows.

Let

$\displaystyle H_{n}(t)$
$\displaystyle=Y_{n}(t)-\int_{0}^{t}Y_{n}(t-s)\,dF(s)\,,$ (3.1) $\displaystyle
X^{(0)}_{n}(t)$
$\displaystyle=\frac{\sqrt{n}}{b_{n}}\,\bigl{(}\frac{1}{n}\,Q_{n}^{(0)}(t)-(1-F_{0}(t))\bigr{)}\,,$
(3.2) $\displaystyle U_{n}(x,t)$
$\displaystyle=\frac{1}{b_{n}\sqrt{n}}\,\sum_{i=1}^{{{\hat{A}}_{n}}(t)}\bigl{(}\,\mathbf{1}_{\\{\eta_{i}\leq
x\\}}\,-F(x)\bigr{)}$ (3.3) and $\displaystyle\Theta_{n}(t)$
$\displaystyle=-\int_{\mathbb{R}_{+}^{2}}\mathbf{1}_{\\{s+x\leq
t\\}}\,dU_{n}(x,s)\,.$ (3.4)

As, owing to (2.2),

$1-F_{0}(t)+\mu t-\mu\int_{0}^{t}(t-s)dF(s)=1\,,$ (3.5)

(2.4) implies that

$\displaystyle X_{n}(t)$
$\displaystyle=(1-F(t))X_{n}(0)^{+}+X_{n}^{(0)}(t)+\int_{0}^{t}X_{n}(t-s)^{+}\,dF(s)+H_{n}(t)+\Theta_{n}(t)\,.$
(3.6)

The equation in (2.14) is written in a similar way: introducing

$\displaystyle H(t)$ $\displaystyle=Y(t)-\int_{0}^{t}Y(t-s)\,dF(s)\,,$ (3.7)
$\displaystyle X^{(0)}(t)$
$\displaystyle=W^{0}(F_{0}(t))-(1-F_{0}(t))X(0)^{-}\,,$ (3.8) $\displaystyle
U(x,t)$ $\displaystyle=K(F(x),\mu t)$ (3.9) and $\displaystyle\Theta(t)$
$\displaystyle=-\int_{\mathbb{R}_{+}^{2}}\mathbf{1}_{\\{s+x\leq
t\\}}\,dU(x,s)=-\int_{\mathbb{R}_{+}^{2}}\mathbf{1}_{\\{s+x\leq
t\\}}\,\dot{K}(F(x),\mu s)\,dF(x)\,\mu ds$ (3.10)

yield

$X(t)=(1-F(t))X(0)^{+}+X^{(0)}(t)+\int_{0}^{t}X(t-s)^{+}\,dF(s)+H(t)+\Theta(t)\,.$
(3.11)

It will be proved that the equations (3.6) LD converge termwise to the
equation (3.11), as $n\to\infty$ . More specifically, the following theorem
will be proved. Introduce $H=(H(t),\,t\in\mathbb{R}_{+})$,
$H_{n}=(H_{n}(t),\,t\in\mathbb{R}_{+})$,
$X^{(0)}=(X^{(0)}(t),\,t\in\mathbb{R}_{+})$,
$X_{n}^{(0)}=(X_{n}^{(0)}(t),\,t\in\mathbb{R}_{+})$,
$U=((U(x,t)\,,x\in\mathbb{R}_{+})\,,t\in\mathbb{R}_{+})$ ,
$U_{n}=((U_{n}(x,t)\,,x\in\mathbb{R}_{+})\,,t\in\mathbb{R}_{+})$ ,
$\Theta=(\Theta(t)\,,t\in\mathbb{R}_{+})$ and
$\Theta_{n}=(\Theta_{n}(t),\,t\in\mathbb{R}_{+})$ .

######  Theorem 3.1.

As $n\to\infty$, the sequence $(X_{n}(0),X_{n}^{(0)},H_{n},\Theta_{n})$ LD
converges in distribution at rate $b_{n}^{2}$ in
$\mathbb{R}\times\mathbb{D}(\mathbb{R}_{+},\mathbb{R})^{3}$ to
$(X(0),X^{(0)},H,\Theta)$ .

The LD convergence of $H_{n}$ and $X_{n}^{(0)}$ to their respective idempotent
counterparts, $H$ and $X^{(0)}$ , is fairly straightforward. The proof of the
LD convergence of $\Theta_{n}$ to $\Theta$ is more involved. The key is LD
convergence of $U_{n}$ to $U$ . Since $U_{n}$ is, essentially, a time–changed
sequential empirical process, one needs a moderate deviation analogue of the
weak convergence of the sequential empirical process to the Kiefer process,
see, e.g., Krichagina and Puhalskii [18]. That result is contained in Lemma
3.1 and may be of interest in its own right. Another important element of the
proof is super–exponential convergence in probability of $\hat{A}_{n}(t)/n$ to
$\mu t$ , which is the subject of Lemma 3.2. The process $\Theta_{n}$ is then
written as

$\Theta_{n}(t)=J_{n}(t)-M_{n}(t)\,,$ (3.12)

where

$\displaystyle J_{n}(t)$
$\displaystyle=\int_{0}^{t}\frac{U_{n}(t-x,x)}{1-F(x)}\,dF(x)$ (3.13)

and $M_{n}=(M_{n}(t)\,,t\in\mathbb{R}_{+})$ is a martingale with respect to a
suitable filtration. By the continuous mapping principle, with $U_{n}$ LD
converging to $U$ , $J_{n}$ LD converges to $J=(J(t)\,,t\in\mathbb{R}_{+})$ ,
where

$J(t)=\int_{0}^{t}\frac{U(t-x,x)}{1-F(x)}\,dF(x)\,.$ (3.14)

It is proved also that the sequence $M_{n}$ LD converges to a certain
idempotent process $M$ . All the convergences holding jointly, the result
follows.

The plan is being implemented next. Let random variables $\zeta_{i}$ be
independent and uniform on $[0,1]$ . Define a centred and normalised
sequential empirical process by

$\displaystyle K_{n}(x,t)=\frac{1}{b_{n}\sqrt{n}}\,\sum_{i=1}^{\lfloor
nt\rfloor}\bigl{(}\,\mathbf{1}_{\\{\zeta_{i}\leq x\\}}\,-x\bigr{)}$ (3.15) and
let $\displaystyle B_{n}(x,t)=\frac{1}{b_{n}\sqrt{n}}\sum_{i=1}^{\lfloor
nt\rfloor}\bigl{(}\,\mathbf{1}_{\\{\zeta_{i}\leq
x\\}}\,-\int_{0}^{x\wedge\zeta_{i}}\frac{dy}{1-y}\bigr{)}\,,$ (3.16)

where $x\in[0,1]$ . It is a simple matter to check that

$K_{n}(x,t)=-\int_{0}^{x}\frac{K_{n}(y,t)}{1-y}\,dy+B_{n}(x,t)\,,\;\;t\in\mathbb{R}_{+}\,,x\in[0,1]\,.$
(3.17)

Let $K_{n}=\bigl{(}(K_{n}(x,t),\,x\in[0,1]),t\in\mathbb{R}_{+}\bigr{)}$ and
$B_{n}=\bigl{(}(B_{n}(x,t),\,x\in[0,1]),t\in\mathbb{R}_{+}\bigr{)}$ . Let
$K=((K(x,t)\,,x\in[0,1])\,,t\in\mathbb{R}_{+})$ represent a Kiefer idempotent
process and let $B=((B(x,t)\,,x\in[0,1])\,,t\in\mathbb{R}_{+})$ represent a
Brownian sheet idempotent process, both being canonical coordinate processes
on $\mathbb{D}(\mathbb{R}_{+},\mathbb{D}([0,1],\mathbb{R}^{2}))$ endowed with
deviability $\Pi$ such that

$K(x,t)=-\int_{0}^{x}\frac{K(y,t)}{1-y}\,dy+B(x,t)\,,\;\;t\in\mathbb{R}_{+}\,,x\in[0,1]\,.$
(3.18)

(That the solution of (3.18) is a Kiefer idempotent process follows from Lemma
A.1.)

######  Lemma 3.1.

If the convergence in (2.12) holds, then the sequence of processes
$(K_{n},B_{n})$ LD converges at rate $b_{n}^{2}$ in
$\mathbb{D}(\mathbb{R}_{+},\mathbb{D}([0,1],\mathbb{R}^{2}))$ to the
idempotent processes $(K,B)$ .

###### Proof.

The proof is modelled on the proof of Lemma 3.1 in Krichagina and Puhalskii
[18], see also Jacod and Shiryaev [15, Chapter IX, §4c]. Given
$t\in\mathbb{R}_{+}$ , let
$K_{n}(t)=(K_{n}(x,t),\,x\in[0,1])\in\mathbb{D}([0,1],\mathbb{R}_{+})$ ,
$B_{n}(t)=(B_{n}(x,t),\,x\in[0,1])\in\mathbb{D}([0,1],\mathbb{R}_{+})$ ,
$K(t)=(K(x,t),\,x\in[0,1])\in\mathbb{D}([0,1],\mathbb{R}_{+})$ and
$B(t)=(B(x,t),\,x\in[0,1])\in\mathbb{D}([0,1],\mathbb{R}_{+})\,.$ It is
proved, first, that, for $0\leq t_{1}<t_{2}<\ldots<t_{k}$, the sequence
$(K_{n}(t_{1}),B_{n}(t_{1})),(K_{n}(t_{2}),B_{n}(t_{2})),\ldots,(K_{n}(t_{k}),B_{n}(t_{k}))$
LD converges to
$(K(t_{1}),B(t_{1})),(K(t_{2}),B(t_{2})),\ldots,(K(t_{k}),B(t_{k})))$ in
$\mathbb{D}([0,1],\mathbb{R}^{2})^{k}$ , as $n\to\infty$ . Since both the
stochastic processes $((K_{n}(t),B_{n}(t))\,,t\in\mathbb{R}_{+})$ and the
idempotent process $((K(t),B(t))\,,t\in\mathbb{R}_{+})$ have independent
increments in $t$ , it suffices to prove convergence of one–dimensional
distributions, so, $((K_{n}(x,t),B_{n}(x,t))\,,x\in[0,1])$ and
$((K(x,t),B(x,t))\,,x\in[0,1])$ are worked with, with $t$ being held fixed.
The process $(B_{n}(x,t)\,,x\in[0,1])$ is a martingale with respect to the
natural filtration with the jump measure

$\mu^{n,B}([0,x],\Gamma)=\,\mathbf{1}_{\\{1/(b_{n}\sqrt{n})\in\Gamma\\}}\,\sum_{i=1}^{\lfloor
nt\rfloor}\,\mathbf{1}_{\\{\zeta_{i}\leq x\\}}\,\,,$

the predictable jump measure

$\nu^{n,B}([0,x],\Gamma)=\,\mathbf{1}_{\\{1/(b_{n}\sqrt{n})\in\Gamma\\}}\,\sum_{i=1}^{\lfloor
nt\rfloor}\int_{0}^{x\wedge\zeta_{i}}\frac{dy}{1-y}$

and the predictable quadratic variation process

$\langle
B_{n}\rangle(x,t)=\int_{0}^{x}\int_{\mathbb{R}}u^{2}\nu^{n,B}(dy,du)=\frac{1}{b_{n}^{2}n}\,\nu^{n,B}([0,x],\\{1/(b_{n}\sqrt{n})\\})=\frac{1}{b_{n}^{2}n}\,\sum_{i=1}^{\lfloor
nt\rfloor}\int_{0}^{x\wedge\zeta_{i}}\frac{dy}{1-y}\\\ =\frac{\lfloor
nt\rfloor}{b_{n}^{2}n}\,x+\frac{1}{b_{n}\sqrt{n}}\,K_{n}(x,t)-\frac{1}{b_{n}\sqrt{n}}\,B_{n}(x,t)\,,$
(3.19)

where $\Gamma\subset\mathbb{R}\setminus\\{0\\}$ . It is shown next that

$\lim_{r\to\infty}\limsup_{n\to\infty}P(\sup_{x\in[0,1]}\lvert
B_{n}(x,t)\rvert>r)^{1/b_{n}^{2}}=0\,.$ (3.20)

An exponential martingale device is applied, which will be used repeatedly.
Since, the process

$\exp\bigl{(}b_{n}^{2}B_{n}(x,t)-\int_{0}^{x}\int_{\mathbb{R}}(e^{b_{n}^{2}u}-1-b_{n}^{2}u)\nu^{n,B}(dy,du)\bigr{)}\\\
=\exp\bigl{(}b_{n}^{2}B_{n}(x,t)-(e^{b_{n}/\sqrt{n}}-1-\frac{b_{n}}{\sqrt{n}})\sum_{i=1}^{\lfloor
nt\rfloor}\int_{0}^{x\wedge\zeta_{i}}\frac{dy}{1-y}\bigr{)}$

is a local martingale with respect to $x$ , see, e.g., Lemma 4.1.1 on p.294 in
Puhalskii [26], for any stopping time $\tau$ ,

$E\exp\bigl{(}b_{n}^{2}B_{n}(\tau,t)-(e^{b_{n}/\sqrt{n}}-1-\frac{b_{n}}{\sqrt{n}})\sum_{i=1}^{\lfloor
nt\rfloor}\int_{0}^{\tau\wedge\zeta_{i}}\frac{dy}{1-y}\bigr{)}\leq 1\,.$
(3.21)

Lemma 3.2.6 on p.282 in Puhalskii [26] implies that, for $r>0$ and $\gamma>0$
,

$P(\sup_{x\in[0,1]}e^{b_{n}^{2}B_{n}(x,t)}\geq e^{b_{n}^{2}r})\leq
e^{b_{n}^{2}(\gamma-r)}+P\bigl{(}\exp\bigl{(}(e^{b_{n}/\sqrt{n}}-1-\frac{b_{n}}{\sqrt{n}})\sum_{i=1}^{\lfloor
nt\rfloor}\int_{0}^{\zeta_{i}}\frac{dy}{1-y}\bigr{)}\geq
e^{b_{n}^{2}\gamma}\bigr{)}\\\ \leq
e^{b_{n}^{2}(\gamma-r)}+e^{-b_{n}^{2}\gamma}E\bigl{(}\exp\bigl{(}(e^{b_{n}/\sqrt{n}}-1-\frac{b_{n}}{\sqrt{n}})\sum_{i=1}^{\lfloor
nt\rfloor}\int_{0}^{\zeta_{i}}\frac{dy}{1-y}\bigr{)}\bigr{)}\\\ \\\
=e^{b_{n}^{2}(\gamma-r)}+e^{-b_{n}^{2}\gamma}\bigl{(}1-(e^{b_{n}/\sqrt{n}}-1-b_{n}/\sqrt{n})\bigr{)}^{-\lfloor
nt\rfloor}\,,$

with the latter equality holding for all $n$ great enough because
$e^{b_{n}/\sqrt{n}}-1-b_{n}/\sqrt{n}\to 0$ . Hence, assuming that
$e^{b_{n}/\sqrt{n}}-1-b_{n}/\sqrt{n}\leq 1/2$ ,

$P(\sup_{x\in[0,1]}e^{b_{n}^{2}B_{n}(x,t)}\geq
e^{b_{n}^{2}r})^{1/b_{n}^{2}}\leq e^{(\gamma-r)}+e^{-\gamma}2^{\lfloor
nt\rfloor/b_{n}^{2}}\,.$

Since $n/b_{n}^{2}\to\infty$ , it follows that

$\lim_{r\to\infty}\limsup_{n\to\infty}P(\sup_{x\in[0,1]}B_{n}(x,t)\geq
r)^{1/b_{n}^{2}}=0\,.$ (3.22)

A similar convergence holds with $-B_{n}(x,t)$ substituted for $B_{n}(x,t)$ .
The limit in (3.20) has been proved.

It is proved next that

$\lim_{r\to\infty}\limsup_{n\to\infty}P(\sup_{x\in[0,1]}\lvert
K_{n}(x,t)\rvert>r)^{1/b_{n}^{2}}=0\,.$ (3.23)

Since owing to (3.15), $(K_{n}(x,t),x\in[0,1])$ is distributed as
$(-K_{n}(1-x,t),x\in[0,1])$ , it suffices to prove that

$\displaystyle\lim_{r\to\infty}\limsup_{n\to\infty}P(\sup_{x\in[0,1/2]}\lvert
K_{n}(x,t)\rvert>r)^{1/b_{n}^{2}}=0\,.$ (3.24)

By (3.17) and the Gronwall–Bellman inequality,

$\sup_{x\in[0,1/2]}\lvert K_{n}(x,t)\rvert\leq e^{2}\sup_{x\in[0,1/2]}\lvert
B_{n}(x,t)\rvert$ (3.25)

so that (3.23) holds too.

By (3.19), (3.23) and (3.20), for $x\in[0,1]$ ,

$\lim_{n\to\infty}P(\lvert b_{n}^{2}\langle
B_{n}\rangle(x,t)-tx\rvert>\epsilon)^{1/b_{n}^{2}}=0\,.$ (3.26)

If extended past $x=1$ by letting $B_{n}(x,t)=B_{n}(1,t)$ , the process
$(B_{n}(x,t)\,,x\in\mathbb{R}_{+})$ is a square integrable martingale with
predictable quadratic variation process $(\langle B_{n}\rangle(x\wedge
1,t)\,,x\in\mathbb{R}_{+})$ , so, by (3.26) and Theorem 5.4.4 on p.423 in
Puhalskii [26], where one takes $\beta_{\phi}=b_{n}\sqrt{n}$ ,
$\alpha_{\phi}=n$ and $r_{\phi}=b_{n}^{2}$ , the sequence of the extended
processes $B_{n}(t)$ LD converges in $\mathbb{D}(\mathbb{R}_{+},\mathbb{R})$
to the idempotent process $(B(x\wedge 1,t)\,,x\in[0,1])$ . By (3.17) and the
continuous mapping principle, for $0\leq x_{1}\leq\ldots\leq x_{l}<1$ , the
$((K_{n}(x_{i},t),B_{n}(x_{i},t))\,,i\in\\{0,1,2,\ldots,l\\})$ LD converge in
$\mathbb{D}(\mathbb{R}_{+},\mathbb{R}^{l})$ to
$((K(x_{i},t),B(x_{i},t))\,,i\in\\{0,1,2,\ldots,l\\})$ . Since $K_{n}(1,t)=0$
and $K(1,t)=0$ (see Appendix A) , the latter convergence also holds if
$x_{l}=1$ .

It is shown now that the sequence $(K_{n}(x,t)\,,x\in[0,1])$ is
$\mathbb{C}$–exponentially tight of order $b_{n}^{2}$ in
$\mathbb{D}([0,1],\mathbb{R})$ . (The definition and basic properties of
$\mathbb{C}$–exponential tightness are reviewed in Appendix A.) By Theorem
A.3, the convergence in (3.23) needs to be complemented with

$\lim_{\delta\rightarrow
0}\limsup_{n\to\infty}\sup_{x\in[0,1]}P(\sup_{y\in[0,\delta]}|K_{n}(x+y,t)-K_{n}(x,t)|\geq\eta)^{1/b_{n}^{2}}=0\,,$
(3.27)

for arbitrary $\eta>0$ , where $K_{n}(x,t)=0$ when $x\geq 1$ . A similar
argument is used as in the proof of (3.23). Defining
$\overline{K}_{n}(x,t)=-K_{n}(1-x,t)$ , for $x\in[0,1]$ , and
$\overline{K}_{n}(x,t)=0$ for $x\geq 1$ , implies, by (3.15), that

$\sup_{x\in[0,1]}P(\sup_{y\in[0,\delta]}|K_{n}(x+y,t)-K_{n}(x,t)|\geq\eta)\leq\sup_{x\in[0,1/2]}P(\sup_{y\in[0,\delta]}|K_{n}(x+y,t)-K_{n}(x,t)|\geq\eta)\\\
+\sup_{x\in[1/2,1-\delta]}P(\sup_{y\in[0,\delta]}|K_{n}(x+y,t)-K_{n}(x,t)|\geq\eta)+\sup_{x\in[1-\delta,1]}P(\sup_{y\in[0,\delta]}|K_{n}(x+y,t)-K_{n}(x,t)|\geq\eta)\\\
\leq\sup_{x\in[0,1/2]}P(\sup_{y\in[0,\delta]}|K_{n}(x+y,t)-K_{n}(x,t)|\geq\eta)+\sup_{x\in[\delta,1/2]}P(\sup_{y\in[0,\delta]}|K_{n}(1-x+y,t)-K_{n}(1-x,t)|\geq\eta)\\\
+P(\sup_{u\in[1-\delta,1]}|K_{n}(u,t)|\geq\eta/2)\leq\sup_{x\in[0,1/2]}P(\sup_{y\in[0,\delta]}|K_{n}(x+y,t)-K_{n}(x,t)|\geq\eta)\\\
+\sup_{x\in[\delta,1/2]}P(\sup_{y\in[0,\delta]}|\overline{K}_{n}(x-y,t)-\overline{K}_{n}(x,t)|\geq\eta)+P(\sup_{u\in[0,\delta]}|\overline{K}_{n}(u,t)|\geq\eta/2)\\\
\leq\sup_{x\in[0,1/2]}P(\sup_{y\in[0,\delta]}|K_{n}(x+y,t)-K_{n}(x,t)|\geq\eta)+\sup_{x\in[0,1/2]}P(\sup_{y\in[0,\delta]}|\overline{K}_{n}(x+y,t)-\overline{K}_{n}(x,t)|\geq\eta)\\\
+P(\sup_{u\in[0,\delta]}|\overline{K}_{n}(u,t)|\geq\eta/2)\,.$ (3.28)

Since the random variables $\zeta_{i}$ are independent and uniformly
distributed on $[0,1]$ , $\overline{K}_{n}$ has the same finite dimensional
distributions as $K_{n}$ so that

$\sup_{x\in[0,1]}P(\sup_{y\in[0,\delta]}|K_{n}(x+y,t)-K_{n}(x,t)|\geq\eta)\leq
2\sup_{x\in[0,1/2]}P(\sup_{y\in[0,\delta]}|K_{n}(x+y,t)-K_{n}(x,t)|\geq\eta)\\\
+P(\sup_{u\in[0,\delta]}|K_{n}(u,t)|\geq\eta/2)\,.$ (3.29)

Since $x+y\leq 2/3$ , when $x\in[0,1/2]$ and $y\in[0,\delta]$ , provided
$\delta$ is small enough, by (3.17), for $x\in[0,1/2]$ and $\delta$ small
enough,

$\sup_{y\in[0,\delta]}\lvert K_{n}(x+y,t)-K_{n}(x,t)\rvert\leq
3\delta\sup_{u\in[0,1]}\lvert K_{n}(u,t)\rvert+\sup_{y\in[0,\delta]}\lvert
B_{n}(x+y,t)-B_{n}(x,t)\rvert\,.$ (3.30)

Similarly,

$\sup_{x\in[0,\delta]}|K_{n}(x,t)|\leq\frac{\delta}{1-\delta}\,\sup_{x\in[0,1]}\lvert
K_{n}(x,t)\rvert+\sup_{x\in[0,\delta]}|B_{n}(x,t)|\,.$ (3.31)

By (3.29), (3.30), (3.31) and the fact that the $B_{n}$ LD converge to $B$ ,

$\limsup_{n\to\infty}\sup_{x\in[0,1]}P(\sup_{y\in[0,\delta]}|K_{n}(x+y,t)-K_{n}(x,t)|\geq\eta)^{1/b_{n}^{2}}\\\
\leq\limsup_{n\to\infty}P(3\delta\sup_{u\in[0,1]}\lvert
K_{n}(u,t)\rvert\geq\eta/2)^{1/b_{n}^{2}}+\limsup_{n\to\infty}P(\frac{\delta}{1-\delta}\,\sup_{u\in[0,1]}\lvert
K_{n}(u,t)\rvert\geq\eta/4)^{1/b_{n}^{2}}\\\
+\Pi(\sup_{x\in[0,1/2]}\sup_{y\in[0,\delta]}\lvert
B(x+y,t)-B(x,t)\rvert\geq\eta/2)+\Pi(\sup_{u\in[0,\delta]}|B(u,t)|\geq\eta/4)\,.$
(3.32)

By (3.23),

$\lim_{\delta\to
0}\limsup_{n\to\infty}\sup_{x\in[0,1]}P(\sup_{y\in[0,\delta]}|K_{n}(x+y,t)-K_{n}(x,t)|\geq\eta)^{1/b_{n}^{2}}\\\
\leq\lim_{\delta\to 0}\Pi(\sup_{x\in[0,1/2]}\sup_{y\in[0,\delta]}\lvert
B(x+y,t)-B(x,t)\rvert\geq\eta/2)+\lim_{\delta\to
0}\Pi(\sup_{u\in[0,\delta]}|B(u,t)|\geq\eta/4)\,.$ (3.33)

Since the collections of sets
$\\{b\in\mathbb{C}(\mathbb{R}_{+},\mathbb{C}([0,1],\mathbb{R})):\,\sup_{x\in[0,1/2]}\sup_{y\in[0,\delta]}\lvert
b(x+y,t)-b(x,t)\rvert\geq\eta/2\\}$ and
$\\{b\in\mathbb{C}(\mathbb{R}_{+},\mathbb{C}([0,1],\mathbb{R})):\,\sup_{x\in[0,\delta]}\lvert
b(x,t)\rvert\geq\eta/4\\}$ are nested collections of closed sets as
$\delta\downarrow 0$ , the limit on the right of (3.33), is (see Appendix A)

$\Pi(\sup_{x\in[0,1/2]}\sup_{y\in[0,0]}\lvert
B(x+y,t)-B(x,t)\rvert\geq\eta/2)+\Pi(\sup_{u\in[0,0]}|B(u,t)|\geq\eta/4)=0\,,$

which concludes the proof of (3.27).

Since the sequence $((K_{n}(x,t),B_{n}(x,t))\,,x\in[0,1])$ LD converges to
$((K(x,t),B(x,t))\,,x\in[0,1])$ in the sense of finite–dimensional
distributions and is $\mathbb{C}$–exponentially tight, the LD convergence
holds in $\mathbb{D}([0,1],\mathbb{R}^{2})$ , see Theorem A.2. It has thus
been proved that the sequence
$\bigl{(}((K_{n}(x,t_{1}),B_{n}(x,t_{1})),\,x\in[0,1]),\ldots,((K_{n}(x,t_{l}),B_{n}(x,t_{l})),\,x\in[0,1])\bigr{)}$
LD converges in $\mathbb{D}([0,1],\mathbb{R}^{2})^{l}$ to
$\bigl{(}((K(x,t_{1}),B(x,t_{1})),\,x\in[0,1]),\ldots,((K(x,t_{l}),B(x,t_{l})),\,x\in[0,1])\bigr{)}$
, for all $t_{1}\leq t_{2}\leq\ldots\leq t_{l}$ . The proof of the lemma will
be complete if the sequence
$\bigl{(}((K_{n}(x,t),B_{n}(x,t)),x\in[0,1])\,,t\in\mathbb{R}_{+}\bigr{)}$ is
proved to be $\mathbb{C}$–exponentially tight of order $b_{n}^{2}$ in
$\mathbb{D}(\mathbb{R}_{+},\mathbb{D}([0,1],\mathbb{R}^{2}))$ . The definition
of being exponentially tight implies that it is sufficient to prove that each
of the sequences $\\{K_{n},n\geq 1\\}$ and $\\{B_{n},n\geq 1\\}$ is
$\mathbb{C}$–exponentially tight of order $b_{n}^{2}$ in
$\mathbb{D}(\mathbb{R}_{+},\mathbb{D}([0,1],\mathbb{R}))$. By (3.23) and
Theorem A.3, the $\mathbb{C}$–exponential tightness of $\\{K_{n},n\geq 1\\}$
would follow if, for all $L>0$ and $\eta>0$ ,

$\displaystyle\lim_{\delta\rightarrow
0}\limsup_{n\to\infty}\sup_{s\in[0,L]}P(\sup_{t\in[0,\delta]}\sup_{x\in[0,1]}|K_{n}(x,s+t)-K_{n}(x,s)|\geq\eta)^{1/b_{n}^{2}}=0.$
(3.34)

Since, in analogy with the reasoning in (3.28),

$P(\sup_{t\in[0,\delta]}\sup_{x\in[0,1]}|K_{n}(x,t+s)-K_{n}(x,s)|\geq\eta)\leq
P(\sup_{t\in[0,\delta]}\sup_{x\in[0,1/2]}|K_{n}(x,t+s)-K_{n}(x,s)|\geq\eta)\\\
+P(\sup_{t\in[0,\delta]}\sup_{x\in[1/2,1]}|K_{n}(x,t+s)-K_{n}(x,s)|\geq\eta)=P(\sup_{t\in[0,\delta]}\sup_{x\in[0,1/2]}|K_{n}(x,t+s)-K_{n}(x,s)|\geq\eta)\\\
+P(\sup_{t\in[0,\delta]}\sup_{x\in[0,1/2]}|\overline{K}_{n}(x,t+s)-\overline{K}_{n}(x,s)|\geq\eta)=2P(\sup_{t\in[0,\delta]}\sup_{x\in[0,1/2]}|K_{n}(x,t+s)-K_{n}(x,s)|\geq\eta)\,,$

(3.34) is implied by

$\displaystyle\lim_{\delta\rightarrow
0}\limsup_{n}\sup_{s\in[0,L]}P(\sup_{t\in[0,\delta]}\sup_{x\in[0,1/2]}|K_{n}(x,t+s)-K_{n}(x,s)|\geq\eta)^{1/b_{n}^{2}}=0.$
(3.35)

With $x$ being held fixed, the process
$(K_{n}(x,t+s)-K_{n}(x,s),t\in\mathbb{R}_{+})$ is a locally square–integrable
martingale, so
$(\sup_{x\in[0,1/2]}(K_{n}(x,t+s)-K_{n}(x,s)),t\in\mathbb{R}_{+})$, is a
submartingale, hence by Doob’s inequality, Liptser and Shiryaev [19, Theorem
3.2, p.60],

$\displaystyle
P(\sup_{t\in[0,\delta]}\sup_{x\in[0,1/2]}|K_{n}(x,t+s)-K_{n}(x,s)|\geq\eta)\leq\frac{1}{\eta^{2b_{n}^{2}}}\,E\sup_{x\in[0,1/2]}(K_{n}(x,s+\delta)-K_{n}(x,s))^{2b_{n}^{2}}.$
(3.36)

As noted earlier, with $t$ being held fixed, the process
$(B_{n}(x,t),x\in[0,1])$ is a square–integrable martingale, Jacod and Shiryaev
[15, Chapter II, §3c]. Equation (3.17) yields, by the Gronwall–Bellman
inequality, as in (3.25),

$\displaystyle\sup_{x\in[0,1/2]}|K_{n}(x,s+\delta)-K_{n}(x,s)|\leq
e^{2}\sup_{x\in[0,1]}|B_{n}(x,s+\delta)-B_{n}(x,s)|\,.$ (3.37)

Since $(B_{n}(x,s+\delta)-B_{n}(x,s),x\in[0,1])$ is a square–integrable
martingale, by another application of Doob’s inequality, Jacod and Shiryaev
[15, Theorem I.1.43], Liptser and Shiryaev [20, Theorem I.9.2], as well as by
Jensen’s inequality,

$\displaystyle
E\sup_{x\in[0,1]}(B_{n}(x,s+\delta)-B_{n}(x,s))^{2b_{n}^{2}}\leq\Bigl{(}\frac{2b_{n}^{2}}{2b_{n}^{2}-1}\Bigr{)}^{2b_{n}^{2}}E(B_{n}(1,s+\delta)-B_{n}(1,s))^{2b_{n}^{2}}\,.$
(3.38)

By (3.16) and the bound (5.6) in the proof of Theorem 19, Chapter III, §5 in
Petrov [22],

$E(B_{n}(1,s+\delta)-B_{n}(1,s))^{2b_{n}^{2}}\leq(b_{n}\sqrt{n})^{-2b_{n}^{2}}\bigl{(}(b_{n}^{2}+1)^{2b_{n}^{2}}n\delta
E(1+\ln\zeta_{1})^{2b_{n}^{2}}\\\
+2b_{n}^{2}(b_{n}^{2}+1)^{b_{n}^{2}}e^{2b_{n}^{2}}(n\delta)^{b_{n}^{2}}\bigl{(}E(1+\ln\zeta_{1})^{2}\bigr{)}^{b_{n}^{2}}\bigr{)}$
(3.39)

so that

$\limsup_{n\to\infty}(E(B_{n}(1,s+\delta)-B_{n}(1,s))^{2b_{n}^{2}})^{1/b_{n}^{2}}\leq\limsup_{n\to\infty}(b_{n}\sqrt{n})^{-2}\bigl{(}(b_{n}^{2}+1)^{2}n^{1/b_{n}^{2}}(E(1+\ln\zeta_{1})^{2b_{n}^{2}})^{1/b_{n}^{2}}\\\
+(b_{n}^{2}+1)e^{2}n\delta
E(1+\ln\zeta_{1})^{2}\bigr{)}\leq\limsup_{n\to\infty}(b_{n}\sqrt{n})^{-2}\bigl{(}(b_{n}^{2}+1)^{2}\bigl{(}(2b_{n}^{2})!\bigr{)}^{1/b_{n}^{2}}+(b_{n}^{2}+1)e^{2}n\delta
E(1+\ln\zeta_{1})^{2}\bigr{)}$

which implies, on recalling that $b_{n}^{3}n^{1/b_{n}^{2}-1/2}\to 0$ , that

$\lim_{\delta\to
0}\limsup_{n\to\infty}\sup_{s\in[0,L]}\Bigl{(}E(B_{n}(1,s+\delta)-B_{n}(1,s))^{2b_{n}^{2}}\Bigr{)}^{1/b_{n}^{2}}=0\,.$

Recalling (3.36), (3.37) and (3.38) obtains (3.35). The proof of the
$\mathbb{C}$–exponential tightness of $B_{n}$ is similar. (It is actually
simpler.) Finally, (3.18) is obtained by letting $n\to\infty$ in (3.17).

∎

The proof of Theorem 3.1 is in order now, so, the hypotheses of Theorem 2.2
are assumed. Since the $F(\eta_{i})$ are iid and uniform on $[0,1]$ , in view
of (3.3) and (3.15), it may be assumed that

$U_{n}(x,t)=K_{n}(F(x),\frac{\hat{A}_{n}(t)}{n})\,.$ (3.40)

On recalling (3.9), thanks to the large deviation version of Slutsky’s
theorem, see Theorem A.1, the LD convergence of $U_{n}$ to $U$ in
$\mathbb{D}(\mathbb{R}_{+},\mathbb{D}(\mathbb{R}_{+},\mathbb{R}))$ is a
consequence of Lemma 3.1 and the following result.

######  Lemma 3.2.

For arbitrary $\epsilon>0$ and $t>0$ ,

$\lim_{n\to\infty}P\bigl{(}\sup_{s\in[0,t]}\lvert\frac{\hat{A}_{n}(s)}{n}-\mu
s\rvert>\epsilon\bigr{)}^{1/b_{n}^{2}}=0\,.$

First, the groundwork is laid and gaps left in the earlier presentation are
filled in. Let

$L_{n}(x,t)=\frac{1}{b_{n}\sqrt{n}}\,\sum_{i=1}^{{{\hat{A}}_{n}}(t)}\Bigl{(}\,\mathbf{1}_{\\{\eta_{i}\leq
x\\}}\,-\int_{0}^{\eta_{i}\wedge x}\frac{dF(u)}{1-F(u)}\Bigr{)}\,.$ (3.41)

By (3.16), it may be assumed that

$L_{n}(x,t)=B_{n}(F(x),\frac{\hat{A}_{n}(t)}{n})\,.$ (3.42)

By (3.3),

$U_{n}(x,t)=-\int_{0}^{x}\frac{U_{n}(y,t)}{1-F(y)}\,dF(y)+L_{n}(x,t)$ (3.43)

so that, owing to (3.4), (3.12) holds with $J_{n}(t)$ defined by (3.13) and
with

$\displaystyle M_{n}(t)$
$\displaystyle=\int_{\mathbb{R}_{+}^{2}}\,\mathbf{1}_{\\{s+x\leq
t\\}}\,\,dL_{n}(x,s)\,.$ (3.44)

Two results on what could be called ”asymptotic boundedness in probability of
order $b_{n}^{2}$” are proved next. More specifically, it is proved that, for
$t>0$ ,

$\displaystyle\lim_{r\to\infty}\limsup_{n\to\infty}P(\sup_{s\in[0,t]}\lvert
U_{n}(s,t-s)\rvert>r)^{1/b_{n}^{2}}=0$ (3.45) and
$\displaystyle\lim_{r\to\infty}\limsup_{n\to\infty}P(\sup_{s\in[0,t]}\lvert
M_{n}(s)\rvert>r)^{1/b_{n}^{2}}=0\,.$ (3.46)

###### Proof of (3.45).

Note that

$\lim_{r\to\infty}\limsup_{n\to\infty}P\bigl{(}\frac{A_{n}(t)}{n}>r\bigr{)}^{1/b_{n}^{2}}=0\,,$
(3.47)

which is a consequence of the LD convergence at rate $b_{n}^{2}$ of the
$Y_{n}$ to $Y$ (see (2.13)). Similarly, the LD convergence of $X_{n}(0)$ to
$X(0)$ implies that

$\lim_{r\to\infty}\limsup_{n\to\infty}P\bigl{(}\sup_{s\in[0,t]}\frac{Q_{n}(0)}{n}>r\bigr{)}^{1/b_{n}^{2}}=0\,.$
(3.48)

By (2.3), ${{\hat{A}}_{n}}(t)\leq(Q_{n}(0)-n)^{+}+A_{n}(t)$ so that (3.47) and
(3.48) imply that

$\lim_{L\to\infty}\limsup_{n\to\infty}P\bigl{(}\frac{\hat{A}_{n}(t)}{n}>L\bigr{)}^{1/b_{n}^{2}}=0\,.$
(3.49)

By (3.40),

$P(\sup_{s\in[0,t]}\lvert U_{n}(s,t-s)\rvert>r)\leq
P(\frac{\hat{A}_{n}(t)}{n}>L)+P(\sup_{\begin{subarray}{c}s\in[0,L],\\\
x\in[0,1]\end{subarray}}\lvert K_{n}(x,s)\rvert>r)\,.$ (3.50)

By the LD convergence of $K_{n}$ to $K$ in Lemma 3.1 and the trajectories of
$K$ being continuous,

$\lim_{r\to\infty}\limsup_{n\to\infty}P\bigl{(}\sup_{\begin{subarray}{c}s\in[0,L],\\\
x\in[0,1]\end{subarray}}\lvert K_{n}(x,s)\rvert>r)^{1/b_{n}^{2}}=0\,.$

Combined with (3.49) and (3.50), this proves (3.45). ∎

###### Proof of (3.46).

As follows from Lemma 3.1 in Puhalskii and Reed [28], the process
$M_{n}=(M_{n}(t),\,t\in\mathbb{R}_{+})$ is a local martingale with respect to
filtration $\mathbf{G}_{n}$ defined as follows. For $t\in\mathbb{R}_{+}$, let
$\hat{\mathcal{G}}_{n}(t)$ denote the complete $\sigma$-algebra generated by
the random variables $\,\mathbf{1}_{\\{\hat{\tau}_{n,i}\leq
s\\}}\,\,\mathbf{1}_{\\{\eta_{i}\leq x\\}}\,$, where $x+s\leq t$ and
$i\in\mathbb{N}$, and by the ${{\hat{A}}_{n}}(s)$ (or, equivalently, by the
$\,\mathbf{1}_{\\{\hat{\tau}_{n,i}\leq s\\}}\,$ for $i\in\mathbb{N}$), where
$s\leq t$. Define
$\mathcal{G}_{n}(t)=\cap_{\epsilon>0}\hat{\mathcal{G}}_{n}(t+\epsilon)$ and
$\mathbf{G}_{n}=(\mathcal{G}_{n}(t)\,,t\in\mathbb{R}_{+})$ . By (3.41) and
(3.44),

$M_{n}(t)=\frac{1}{b_{n}\sqrt{n}}\,\sum_{i=1}^{{{\hat{A}}_{n}}(t)}\bigl{(}\,\mathbf{1}_{\\{\eta_{i}+\hat{\tau}_{n,i}\leq
t\\}}\,-\int_{0}^{\eta_{i}\wedge(t-\hat{\tau}_{n,i})}\frac{dF(u)}{1-F(u)}\bigr{)}\,.$
(3.51)

Thus, the measure of jumps of $M_{n}$ is

$\mu_{n}([0,t],\Gamma)=\,\mathbf{1}_{\\{1/(b_{n}\sqrt{n})\in\Gamma\\}}\,\sum_{i=1}^{{{\hat{A}}_{n}}(t)}\,\mathbf{1}_{\\{\eta_{i}+\hat{\tau}_{n,i}\leq
t\\}}\,$ (3.52)

and the associated $\mathbf{G}_{n}$–predictable measure of jumps is

$\nu_{n}([0,t],\Gamma)=\,\mathbf{1}_{\\{1/(b_{n}\sqrt{n})\in\Gamma\\}}\,\sum_{i=1}^{{{\hat{A}}_{n}}(t)}\int_{0}^{\eta_{i}\wedge(t-\hat{\tau}_{n,i})}\frac{dF(u)}{1-F(u)}\,.$
(3.53)

Note that it is a continuous process. (For $\hat{A}_{n}$ being predictable,
see Lemma C.1 in Puhalskii and Reed [28].) The associated stochastic cumulant
is (see, e.g., p.293 in Puhalskii [26])

$G_{n}(\alpha,t)=(e^{\alpha/(b_{n}\sqrt{n})}-1-\frac{\alpha}{b_{n}\sqrt{n}})\sum_{i=1}^{{{\hat{A}}_{n}}(t)}\int_{0}^{\eta_{i}\wedge(t-\hat{\tau}_{n,i})}\frac{dF(u)}{1-F(u)}\,.$
(3.54)

By Lemma 4.1.1 on p.294 in Puhalskii [26], the process $(e^{\alpha
M_{n}(t)-G_{n}(\alpha,t)}\,,t\in\mathbb{R}_{+})$ is a local martingale so that
$Ee^{\alpha M_{n}(\tau)-G_{n}(\alpha,\tau)}\leq 1$ , for arbitrary stopping
time $\tau$ . Lemma 3.2.6 on p.282 in Puhalskii [26] implies that, for
$\gamma>0$ ,

$P(\sup_{s\in[0,t]}e^{\alpha b_{n}^{2}M_{n}(s)}\geq e^{\alpha b_{n}^{2}r})\leq
e^{\alpha b_{n}^{2}(\gamma-r)}+P(e^{G_{n}(\alpha b_{n}^{2},t)}\geq e^{\alpha
b_{n}^{2}\gamma})\leq e^{\alpha b_{n}^{2}(\gamma-r)}+P(e^{\hat{G}_{n}(\alpha
b_{n}^{2},t)}\geq e^{\alpha b_{n}^{2}\gamma})\,,$

where

$\hat{G}_{n}(\alpha,t)=(e^{\alpha/(b_{n}\sqrt{n})}-1-\frac{\alpha}{b_{n}\sqrt{n}})\sum_{i=1}^{{{\hat{A}}_{n}}(t)}\int_{0}^{t-\hat{\tau}_{n,i}}\frac{dF(u)}{1-F(u)}\,.$
(3.55)

Hence, for $\alpha>0$ ,

$P(\sup_{s\in[0,t]}M_{n}(s)\geq r)^{1/b_{n}^{2}}\leq
e^{\alpha(\gamma-r)}+P(\hat{G}_{n}(\alpha b_{n}^{2},t)\geq\alpha
b_{n}^{2}\gamma)^{1/b_{n}^{2}}\,.$ (3.56)

On writing

$\hat{G}_{n}(\alpha b_{n}^{2},t)=(e^{\alpha b_{n}/\sqrt{n}}-1-\frac{\alpha
b_{n}}{\sqrt{n}})\int_{0}^{t}\int_{0}^{t-s}\frac{dF(u)}{1-F(u)}\,d\hat{A}_{n}(s)\\\
=(e^{\alpha b_{n}/\sqrt{n}}-1-\frac{\alpha
b_{n}}{\sqrt{n}})\int_{0}^{t}\hat{A}_{n}(t-u)\,\frac{dF(u)}{1-F(u)}$ (3.57)

and noting that $(n/b_{n}^{2})(e^{\alpha b_{n}/\sqrt{n}}-1-\alpha
b_{n}/\sqrt{n})\to\alpha^{2}/2$ , one can see, thanks to (3.49), that

$\lim_{n\to\infty}P(\hat{G}_{n}(\alpha b_{n}^{2},t)\geq\alpha
b_{n}^{2}\gamma)^{1/b_{n}^{2}}=0\,,$

provided $\alpha$ is small enough, which proves that

$\lim_{r\to\infty}\limsup_{n\to\infty}P(\sup_{s\in[0,t]}M_{n}(s)>r)^{1/b_{n}^{2}}=0\,.$

The argument for $\sup_{s\in[0,t]}(-M_{n}(s))$ is similar. ∎

###### Proof of Lemma 3.2.

By (2.3), (2.4), (3.3) and (3.4),

$\frac{1}{n}\,Q_{n}(t)=\bigl{(}\frac{1}{n}\,Q_{n}(0)-1\bigr{)}^{+}(1-F(t))+\frac{1}{n}\,Q_{n}^{(0)}(t)+\frac{1}{n}\,A_{n}(t)-\frac{1}{n}\,\int_{0}^{t}A_{n}(t-s)\,dF(s)\\\
+\frac{1}{n}\,\int_{0}^{t}(Q_{n}(t-s)-n)^{+}\,dF(s)+\frac{b_{n}}{\sqrt{n}}\,\Theta_{n}(t)\,.$
(3.58)

By (3.12), (3.13), (3.45) and (3.46), on recalling that $F(t)<1$ ,

$\lim_{r\to\infty}\limsup_{n\to\infty}P(\sup_{s\in[0,t]}\lvert\Theta_{n}(s)\rvert>r)^{1/b_{n}^{2}}=0\,.$
(3.59)

The LD convergence at rate $b_{n}^{2}$ of $Y_{n}$ to $Y$ implies that, for
$\epsilon>0$ ,

$\lim_{n\to\infty}P\bigl{(}\sup_{s\in[0,t]}\lvert\frac{A_{n}(s)}{n}-\mu
s\rvert>\epsilon\bigr{)}^{1/b_{n}^{2}}=0\,.$ (3.60)

By (2.5), (3.48) and Lemma 3.1,

$\lim_{n\to\infty}P(\sup_{s\in[0,t]}\lvert\frac{1}{n}\,Q_{n}^{(0)}(s)-(1-F_{0}(s))\rvert>\epsilon)^{1/b_{n}^{2}}=0\,.$
(3.61)

Recalling (3.5) and (3.47) implies that

$\lim_{n\to\infty}P(\sup_{s\in[0,t]}\lvert\frac{1}{n}\,Q_{n}^{(0)}(s)+\frac{1}{n}\,A_{n}(s)-\frac{1}{n}\,\int_{0}^{s}A_{n}(s-x)\,dF(x)-1\rvert>\epsilon)^{1/b_{n}^{2}}=0\,.$
(3.62)

Besides, the LD convergence of $X_{n}(0)$ to $X(0)$ implies that (3.48) can be
strengthened as follows:

$\limsup_{n\to\infty}P\bigl{(}\lvert\frac{Q_{n}(0)}{n}-1\rvert>\epsilon\bigr{)}^{1/b_{n}^{2}}=0\,.$
(3.63)

Hence, by (3.58) and (3.59),

$\frac{1}{n}\,Q_{n}(t)-1=\,\int_{0}^{t}(\frac{1}{n}\,Q_{n}(t-s)-1)^{+}\,dF(s)+\theta_{n}(t)\,,$
(3.64)

where

$\lim_{n\to\infty}P(\sup_{s\in[0,t]}\lvert\theta_{n}(s)\rvert>\epsilon)^{1/b_{n}^{2}}=0\,.$

Lemma B.1 in Puhalskii and Reed [28] implies that there exists function $\rho$
which depends on the function $F$ only such that

$\sup_{s\in[0,t]}\lvert\frac{1}{n}\,Q_{n}(s)-1\rvert\leq\rho(t)\sup_{s\in[0,t]}\lvert\theta_{n}(s)\rvert\,.$

Therefore,

$\lim_{n\to\infty}P(\sup_{s\in[0,t]}\lvert\frac{1}{n}\,Q_{n}(s)-1\rvert>\epsilon)^{1/b_{n}^{2}}=0\,.$
(3.65)

When combined with (2.3) and (3.60), this yields the assertion of the lemma. ∎

As mentioned, the LD convergence of $U_{n}$ to $U$ has been proved thereby.
Let
$L_{n}=\bigl{(}(L_{n}(x,t)\,,x\in\mathbb{R}_{+}),t\in\mathbb{R}_{+}\bigr{)}$
and $L=\bigl{(}(L(x,t)\,,x\in\mathbb{R}_{+}),t\in\mathbb{R}_{+}\bigr{)}$ ,
with $L(x,t)=B(F(x),\mu t)$ .

######  Lemma 3.3.

As $n\to\infty$, the sequence $(X_{n}(0),X_{n}^{(0)},U_{n},L_{n})$ LD
converges in distribution in
$\mathbb{R}\times\mathbb{D}(\mathbb{R}_{+},\mathbb{R})\times\mathbb{D}(\mathbb{R}_{+},\mathbb{D}(\mathbb{R}_{+},\mathbb{R}))^{2}$
to $(X(0),X^{(0)},U,L)$ .

###### Proof.

Let

$X^{\prime(0)}_{n}(x,t)=\frac{1}{b_{n}\sqrt{n}}\,\sum_{i=1}^{\lfloor
nt\rfloor}\,(\,\mathbf{1}_{\\{\eta^{(0)}_{i}>x\\}}\,-(1-F_{0}(x)))$ (3.66)

and $X^{\prime(0)}_{n}=(X^{\prime(0)}_{n}(t)\,,t\in\mathbb{R}_{+})$ . By the
hypotheses of Theorem 2.2, $X_{n}(0)$ LD converges to $X(0)$ , by Lemma 3.1
and $F_{0}$ being strictly increasing, $X^{\prime(0)}_{n}$ LD converges to
$((\tilde{K}(F_{0}(x),t)\,,x\in\mathbb{R}_{+})\,,t\in\mathbb{R}_{+})$ , where
$\tilde{K}$ represents a Kiefer idempotent process that is independent of
$(X(0),K,B)$ . Also $(K_{n},B_{n})$ LD converges to $(K,B)$ . By independence
assumptions, these convergences hold jointly, cf. Section A. Since
$Q_{n}(0)/n\to 1$ and $\hat{A}_{n}(t)/n\to\mu t$ super–exponentially in
probability by (3.63) and (3.60), respectively, ”Slutsky’s theorem” (Lemma
A.1) yields joint LD convergence of
$(X_{n}(0),X^{\prime(0)}_{n},K_{n},B_{n},Q_{n}(0)/n,(\hat{A}_{n}(t)/n\,,t\in\mathbb{R}_{+}))$
to
$(X(0),((\tilde{K}(F_{0}(x),t)\,,x\in\mathbb{R}_{+})\,,t\in\mathbb{R}_{+}),K,B,1,(\mu
t\,,t\in\mathbb{R}_{+}))$ . It is also noteworthy that, by (2.5), (3.2) and
(3.66),

$X^{(0)}_{n}(t)=X^{\prime(0)}_{n}(t,\frac{Q_{n}(0)}{n}\wedge
1)-(1-F_{0}(t))X_{n}(0)^{-}\,.$ (3.67)

In order to deduce the LD convergence of $(X_{n}(0),X_{n}^{(0)},U_{n},L_{n})$
to $(X(0),X^{(0)},U,L)$ , it is left to recall (3.8), (3.40), (3.42), note
that, by Lemma A.1, $(\tilde{K}(t,1)\,,t\in[0,1])$ is a Brownian bridge
idempotent process, and apply the continuous mapping principle, the associated
composition mappings being continuous at continuous limits. (See Whitt [30]
for more background on continuous functions in the Skorohod space context.) ∎

######  Lemma 3.4.

The sequence $\\{\Theta_{n}\,,n\in\mathbb{N}\\}$ is $\mathbb{C}$–exponentially
tight of order $b_{n}^{2}$ in $\mathbb{D}(\mathbb{R}_{+},\mathbb{R})$ .

###### Proof.

By Lemma 3.3, (3.13), (3.14) and the continuous mapping principle, $J_{n}$ LD
converges to $J$ , so, the sequence $J_{n}$ is $\mathbb{C}$–exponentially
tight. By (3.12) it remains to check that the sequence $M_{n}$ is
$\mathbb{C}$–exponentially tight, which, according to Theorem A.3, is implied
by the following convergences:

$\displaystyle\lim_{r\to\infty}\limsup_{n\to\infty}P(\sup_{s\in[0,t]}\lvert
M_{n}(s)\rvert>r)^{1/b_{n}^{2}}=0$ and $\displaystyle\lim_{\delta\to
0}\limsup_{n\to\infty}\sup_{s\in[0,t]}P(\sup_{s^{\prime}\in[0,\delta]}\lvert
M_{n}(s^{\prime}+s)-M_{n}(s)\rvert>\epsilon)^{1/b_{n}^{2}}=0\,,$ (3.68)

where $t>0$ and $\epsilon>0$ . The former convergence has already been proved,
see (3.46). The proof of (3.68) proceeds along similar lines. Since, with
$\alpha\in\mathbb{R}$ , the process
$(e^{\alpha(M_{n}(s+s^{\prime})-M_{n}(s))-(G_{n}(\alpha,s+s^{\prime})-G_{n}(\alpha,s))}\,,s^{\prime}\in\mathbb{R}_{+})$
is a local martingale so that, for arbitrary stopping time $\tau$ ,
$Ee^{\alpha(M_{n}(\tau+s)-M_{n}(s))-(G_{n}(\alpha,\tau+s)-G_{n}(\alpha,s))}\leq
1$ , by Lemma 3.2.6 on p.282 in Puhalskii [26], for arbitrary $\gamma>0$ , in
analogy with (3.56),

$P(\sup_{s^{\prime}\in[0,\delta]}(M_{n}(s^{\prime})-M_{n}(s))\geq\epsilon)^{1/b_{n}^{2}}\leq
e^{\alpha(\gamma-\epsilon)}+P(\hat{G}_{n}(\alpha
b_{n}^{2},s+\delta)-\hat{G}_{n}(\alpha b_{n}^{2},s)\geq\alpha
b_{n}^{2}\gamma)^{1/b_{n}^{2}}\,.$

By (3.57) and Lemma 3.2,

$\frac{1}{b_{n}^{2}}\,\hat{G}_{n}(\alpha
b_{n}^{2},t)\to\frac{\alpha^{2}}{2}\,\mu\int_{0}^{t}(t-u)\frac{dF(u)}{1-F(u)}$
(3.69)

super–exponentially in probability at rate $b_{n}^{2}$ . The latter
super–exponential convergence in probability being locally uniform in $t$ , as
the limit is a monotonic continuous function starting at $0$ , implies that,
for $\delta$ small enough, depending on $\alpha$ ,

$\limsup_{n\to\infty}\sup_{s\in[0,t]}P(\sup_{s^{\prime}\in[0,\delta]}(M_{n}(s^{\prime}+s)-M_{n}(s))\geq\epsilon)^{1/b_{n}^{2}}\leq
e^{\alpha(\gamma-\epsilon)}\,.$

Now, one chooses $\gamma<\epsilon$ and sends $\alpha\to\infty$ . ∎

######  Lemma 3.5.

As $n\to\infty$, the sequence
$(X_{n}(0),X_{n}^{(0)},H_{n},J_{n},L_{n},U_{n},\Theta_{n})$ LD converges in
distribution in
$\mathbb{R}\times\mathbb{D}(\mathbb{R}_{+},\mathbb{R})^{3}\times\mathbb{D}(\mathbb{R}_{+},\mathbb{D}(\mathbb{R}_{+},\mathbb{R}))^{2}\times\mathbb{D}(\mathbb{R}_{+},\mathbb{R})$
to $(X(0),X^{(0)},H,J,L,U,\Theta)$ .

###### Proof.

Let

$\Theta^{(l)}_{n}(t)=-\int_{\mathbb{R}_{+}^{2}}I_{t}^{(l)}(x,s)dU_{n}(x,s)\,,$
(3.70)

where

$I_{t}^{(l)}(x,s)=\sum_{i=1}^{\infty}\,\mathbf{1}_{\\{s\in(s^{(l)}_{i-1},s^{(l)}_{i}]\\}}\,\,\mathbf{1}_{\\{x\in[0,t-s^{(l)}_{i-1}]\\}}\,\,,$
(3.71)

for some $0=s^{(l)}_{0}<s^{(l)}_{1}<\ldots$ such that $s^{(l)}_{i}\to\infty$ ,
as $i\to\infty$ , and $\sup_{i\geq 1}(s^{(l)}_{i}-s^{(l)}_{i-1})\to 0$ , as
$l\to\infty$ , and, accordingly,

$\int_{\mathbb{R}_{+}^{2}}I_{t}^{(l)}(x,s)dU_{n}(x,s)=\sum_{i=1}^{\infty}(U_{n}(t-s^{(l)}_{i-1},s^{(l)}_{i})-U_{n}(t-s^{(l)}_{i-1}),s^{(l)}_{i-1})\,\mathbf{1}_{\\{s^{(l)}_{i-1}\leq
t\\}}\,\,.$ (3.72)

Similarly, let

$\Theta^{(l)}(t)=-\int_{\mathbb{R}_{+}^{2}}I_{t}^{(l)}(x,s)\,dU(x,s)=\sum_{i=1}^{\infty}(U(t-s^{(l)}_{i-1},s^{(l)}_{i})-U(t-s^{(l)}_{i-1},s^{(l)}_{i-1}))\,\mathbf{1}_{\\{s^{(l)}_{i-1}\leq
t\\}}\,\,.$ (3.73)

By the LD convergence of $Y_{n}$ to $Y$ in the hypotheses of Theorem 2.2,
Lemma 3.3, (3.1), (3.7), (3.13), (3.14), and the continuous mapping principle,
the sequence $(X_{n}(0),X_{n}^{(0)},H_{n},J_{n},L_{n},U_{n})$ LD converges to
$(X(0),X^{(0)},H,J,L,U)$ . Hence, the sequence
$(X_{n}(0),X_{n}^{(0)},H_{n},J_{n},L_{n},U_{n},\Theta_{n}^{(l)})$ LD converges
to $(X(0),X^{(0)},H,J,L,U,\Theta^{(l)})$ . In addition, by Lemma 3.4, the
sequence $\Theta_{n}$ is $\mathbb{C}$–exponentially tight. Since the
idempotent processes $X^{(0)},H,J,L,U$ are seen to have continuous
trajectories, the sequence $(X_{n}^{(0)},H_{n},J_{n},L_{n},U_{n},\Theta_{n})$
is $\mathbb{C}$–exponentially tight. That a limit point of
$(X_{n}(0),X_{n}^{(0)},H_{n},J_{n},L_{n},U_{n},\Theta_{n})$ is distributed as
$(X(0),X^{(0)},H,J,L,U,\Theta)$ would follow from the LD convergence of
finite–dimensional distributions of
$(X_{n}(0),X_{n}^{(0)},H_{n},J_{n},L_{n},U_{n},\Theta_{n})$ to
finite–dimensional distributions of $(X(0),X^{(0)},H,J,L,U,\Theta)$ . Owing to
(3.9),

$\Theta^{(l)}(t)=-\int_{\mathbb{R}_{+}^{2}}I_{t}^{(l)}(x,s)\dot{K}(F(x),\mu
s)\,dF(x)\,\mu ds\,.$ (3.74)

which implies, by (3.10) and the Cauchy–Schwarz inequality, that
$\Theta^{(l)}\to\Theta$ locally uniformly, as $l\to\infty$ . Since the
sequence $(X_{n}(0),X_{n}^{(0)},H_{n},J_{n},L_{n},U_{n})$ LD converges to
$(X(0),X^{(0)},H,J,L,U)$ , in order to prove the finite–dimensional LD
convergence it suffices to prove that

$\lim_{l\to\infty}\limsup_{n\to\infty}P(\lvert\Theta_{n}^{(l)}(t)-\Theta_{n}(t)\rvert>\epsilon)^{1/b_{n}^{2}}=0\,.$
(3.75)

By (3.4) and (3.70),

$\Theta_{n}(t)-\Theta_{n}^{(l)}(t)=\frac{1}{b_{n}\sqrt{n}}\sum_{i=1}^{\hat{A}_{n}(t)}\sum_{j=1}^{\infty}\,\mathbf{1}_{\\{s^{(l)}_{j-1}\leq
t\\}}\,\,\mathbf{1}_{\\{\hat{\tau}_{n,i}\in(s^{(l)}_{j-1},s^{(l)}_{j}]\\}}\,\bigl{(}\,\mathbf{1}_{\\{\eta_{i}\in(t-\hat{\tau}_{n,i},t-s^{(l)}_{j-1}]\\}}\,-(F(t-s^{(l)}_{j-1})-F(t-\hat{\tau}_{n,i})\bigr{)}\,.$

Let

$\hat{\Theta}_{n}^{(l)}(s,t)=\frac{1}{b_{n}\sqrt{n}}\sum_{i=1}^{\hat{A}_{n}(s)}\sum_{j=1}^{\infty}\,\mathbf{1}_{\\{s^{(l)}_{j-1}\leq
t\\}}\,\,\mathbf{1}_{\\{\hat{\tau}_{n,i}\in(s^{(l)}_{j-1},s^{(l)}_{j}]\\}}\,\bigl{(}\,\mathbf{1}_{\\{\eta_{i}\in(t-\hat{\tau}_{n,i},t-s^{(l)}_{j-1}]\\}}\,-(F(t-s^{(l)}_{j-1})-F(t-\hat{\tau}_{n,i}))\bigr{)}\,.$

Let $\mathcal{F}_{n}(s)$ represent the complete $\sigma$–algebra generated by
the random variables $\hat{\tau}_{n,j}\wedge\hat{\tau}_{n,\hat{A}_{n}(s)+1}$
and $\eta_{j\wedge\hat{A}_{n}(s)}$ , where $j\in\mathbb{N}$ . By part 4 of
Lemma C.1 in Puhalskii and Reed [28], with $t$ held fixed, the process
$(\hat{\Theta}_{n}^{(l)}(s,t)\,,s\in\mathbb{R}_{+})$ is an
$\mathbb{F}_{n}$–locally square integrable martingale. Its measure of jumps is

$\mu_{n}^{(l)}([0,s],\Gamma)=\sum_{i=1}^{\hat{A}_{n}(s)}\sum_{j=1}^{\infty}\,\mathbf{1}_{\\{s^{(l)}_{j-1}\leq
t\\}}\,\,\mathbf{1}_{\\{\hat{\tau}_{n,i}\in(s^{(l)}_{j-1},s^{(l)}_{j}]\\}}\,\bigl{(}\,\mathbf{1}_{\\{(1-(F(t-s^{(l)}_{j-1})-F(t-\hat{\tau}_{n,i}))/(b_{n}\sqrt{n})\in\Gamma\\}}\,\,\mathbf{1}_{\\{\eta_{i}\in(t-\hat{\tau}_{n,i},t-s^{(l)}_{j-1}]\\}}\,\\\
+\,\mathbf{1}_{\\{(-(F(t-s^{(l)}_{j-1})-F(t-\hat{\tau}_{n,i}))/(b_{n}\sqrt{n})\in\Gamma\\}}\,\,\mathbf{1}_{\\{\eta_{i}\not\in(t-\hat{\tau}_{n,i},t-s^{(l)}_{j-1}]\\}}\,\bigr{)}\,,$

where $\Gamma\subset\mathbb{R}\setminus\\{0\\}$ . Accordingly, the
$\mathbb{F}_{n}$–predictable measure of jumps is

$\nu_{n}^{(l)}([0,s],\Gamma)\\\
=\sum_{i=1}^{\hat{A}_{n}(s)}\sum_{j=1}^{\infty}\,\mathbf{1}_{\\{s^{(l)}_{j-1}\leq
t\\}}\,\,\mathbf{1}_{\\{\hat{\tau}_{n,i}\in(s^{(l)}_{j-1},s^{(l)}_{j}]\\}}\,\bigl{(}\,\mathbf{1}_{\\{(1-(F(t-s^{(l)}_{j-1})-F(t-\hat{\tau}_{n,i}))/(b_{n}\sqrt{n})\in\Gamma\\}}\,(F(t-s^{(l)}_{j-1})-F(t-\hat{\tau}_{n,i}))\\\
+\,\mathbf{1}_{\\{-(F(t-s^{(l)}_{j-1})-F(t-\hat{\tau}_{n,i}))/(b_{n}\sqrt{n})\in\Gamma\\}}\,(1-(F(t-s^{(l)}_{j-1})-F(t-\hat{\tau}_{n,i})))\bigr{)}\,.$

For $\alpha\in\mathbb{R}$ , as on p.214 in Puhalskii [26], define the
stochastic cumulant

$G^{(l)}_{n}(\alpha,s)=\int_{0}^{s}\int_{\mathbb{R}}(e^{\alpha x}-1-\alpha
x)\nu^{(l)}_{n}(ds^{\prime},dx)\\\
=\sum_{i=1}^{\hat{A}_{n}(s)}\sum_{j=1}^{\infty}\,\mathbf{1}_{\\{s^{(l)}_{j-1}\leq
t\\}}\,\,\mathbf{1}_{\\{\hat{\tau}_{n,i}\in(s^{(l)}_{j-1},s^{(l)}_{j}]\\}}\,\bigl{(}(e^{\alpha(1-(F(t-s^{(l)}_{j-1})-F(t-\hat{\tau}_{n,i})))/(b_{n}\sqrt{n})}\\\
-1-\frac{\alpha}{b_{n}\sqrt{n}}(1-(F(t-s^{(l)}_{j-1})-F(t-\hat{\tau}_{n,i}))))(F(t-s^{(l)}_{j-1})-F(t-\hat{\tau}_{n,i}))\\\
+(e^{-\alpha(F(t-s^{(l)}_{j-1})-F(t-\hat{\tau}_{n,i}))/(b_{n}\sqrt{n})}-1+\frac{\alpha}{b_{n}\sqrt{n}}(F(t-s^{(l)}_{j-1})-F(t-\hat{\tau}_{n,i})))(1-(F(t-s^{(l)}_{j-1})-F(t-\hat{\tau}_{n,i})))\bigr{)}\,.$
(3.76)

The associated stochastic exponential is defined by

$\mathcal{E}^{(l)}_{n}(\alpha,s)=e^{G^{(l)}_{n}(\alpha,s)}\prod_{0<s^{\prime}\leq
s}(1+\Delta G^{(l)}_{n}(\alpha,s^{\prime}))e^{-\Delta
G^{(l)}_{n}(\alpha,s^{\prime})}\,,$ (3.77)

where $\Delta G^{(l)}_{n}(s^{\prime})$ represents the jump of
$G^{(l)}_{n}(s^{\prime\prime})$ with respect to $s^{\prime\prime}$ evaluated
at $s^{\prime\prime}=s^{\prime}$ and the product is taken over the jumps. By
Lemma 4.1.1 on p.294 in Puhalskii [26], the process $\bigl{(}\exp(\alpha
b_{n}^{2}\hat{\Theta}_{n}^{(l)}(s,t))\mathcal{E}^{(l)}_{n}(\alpha
b_{n}^{2},s)^{-1}\,,s\in\mathbb{R}_{+}\bigr{)}$ is a well defined local
martingale so that, for any stopping time $\tau$ , $E\exp(\alpha
b_{n}^{2}\hat{\Theta}_{n}^{(l)}(\tau,t))\mathcal{E}^{(l)}_{n}(\alpha
b_{n}^{2},\tau)^{-1}\leq 1\,.$ Lemma 3.2.6 on p.282 in Puhalskii [26] and
(3.77) imply that, for $\alpha>0$ and $\gamma>0$ ,

$P(\sup_{s\in[0,t]}\hat{\Theta}_{n}^{(l)}(s,t)\geq\epsilon)\leq e^{\alpha
b_{n}^{2}(\gamma-\epsilon)}+P(\mathcal{E}^{(l)}_{n}(\alpha b_{n}^{2},t)\geq
e^{\alpha b_{n}^{2}\gamma})\leq e^{\alpha
b_{n}^{2}(\gamma-\epsilon)}+P(G^{(l)}_{n}(\alpha b_{n}^{2},t)\geq\alpha
b_{n}^{2}\gamma)\,.$ (3.78)

By (A.13),

$G^{(l)}_{n}(\alpha b_{n}^{2},t)\leq\hat{A}_{n}(t)\bigl{(}\sup_{\lvert
y\rvert\leq 1}(e^{\alpha
b_{n}y/\sqrt{n}}-1-\alpha\frac{b_{n}y}{\sqrt{n}})\sup_{j\in\mathbb{N}}(F(t-s^{(l)}_{j-1})-F(t-s^{(l)}_{j}))\\\
+\sup_{j\in\mathbb{N}}\sup_{y\in[F(t-s^{(l)}_{j}),F(t-s^{(l)}_{j-1})]}(e^{-\alpha
b_{n}y/\sqrt{n}}-1+\alpha\frac{b_{n}}{\sqrt{n}}y)\bigr{)}\,.$

As $n/b_{n}^{2}(e^{\alpha b_{n}y/\sqrt{n}}-1-\alpha
b_{n}y/\sqrt{n})\to\alpha^{2}y^{2}/2$ uniformly on bounded intervals,
$\hat{A}_{n}(t)/n\to\mu t$ super–exponentially, as $n\to\infty$, and
$\sup_{j}(s^{l}_{j}-s^{l}_{j-1})\to 0$ , as $l\to\infty$ , it follows that

$\lim_{l\to\infty}\limsup_{n\to\infty}P(G^{(l)}_{n}(\alpha
b_{n}^{2},t)\geq\alpha b_{n}^{2}\gamma)^{1/b_{n}^{2}}=0\,,$

which implies, thanks to (3.78), that

$\limsup_{l\to\infty}\limsup_{n\to\infty}P(\sup_{s\in[0,t]}\hat{\Theta}_{n}^{(l)}(s,t)\geq\epsilon)^{1/b_{n}^{2}}\leq
e^{\alpha(\gamma-\epsilon)}\,.$

Picking $\gamma<\epsilon$ and sending $\alpha$ to $\infty$ show that the
latter lefthand side equals zero. A similar argument proves the convergence

$\lim_{l\to\infty}\limsup_{n\to\infty}P(\sup_{s\in[0,t]}(-\hat{\Theta}_{n}^{(l)}(s,t))\geq\epsilon)^{1/b_{n}^{2}}=0\,.$

Noting that $\Theta_{n}(t)-\Theta_{n}^{(l)}(t)=\hat{\Theta}_{n}^{(l)}(t,t)$
yields the convergence (3.75). Lemma 3.5 has been proved so that Theorem 3.1
has been proved.

∎

Theorem 3.1 has been proved. In order to obtain the assertion of Theorem 2.2,
note that $-K$ has the same idempotent distribution as $K$ and invoke the
continuous mapping principle in (3.6), which applies due to Lemma B.2 in
Puhalskii and Reed [28]. Part 2 of Theorem 2.2 holds because under its
hypotheses, $Y(t)=\sigma W(t)-\beta\mu t$ . The assertion of Theorem 2.1
follows on observing that $\exp(-I^{Q}(y))$, with
$y\in\mathbb{D}(\mathbb{R}_{+},\mathbb{R})$ , is the deviability density of
the idempotent distribution of $X$ in the statement of Theorem 2.2. To see the
latter, note that the mapping $(w^{0},w,k)\to u$ specified by (2.9) is
continuous when restricted to the set
$\\{(w^{0},w,k):\Pi^{W^{0},W,K}(w^{0},w,k)\geq a\\}$ , where
$\Pi^{W^{0},W,K}(w^{0},w,k)=\Pi^{W^{0}}(w^{0})\Pi^{W}(w)\Pi^{K}(k)$ and
$a\in(0,1]$ , so that $X$ is strictly Luzin on
$\bigl{(}\mathbb{D}([0,1],\mathbb{R})\times\mathbb{D}(\mathbb{R}_{+},\mathbb{R})\times\mathbb{D}(\mathbb{R}_{+},\mathbb{D}(\mathbb{R}_{+},\mathbb{R})),\Pi^{W^{0},W,K}\bigr{)}$
, see Appendix A for the definition and properties of being strictly Luzin.
Therefore,

$\Pi^{X}(u)=\Pi^{W^{0},W,K}(X=u)=\sup_{(w^{0},w,k):\,\eqref{eq:1}\text{
holds}}\Pi^{W^{0}}(w^{0})\Pi^{W}(w)\Pi^{K}(k)\,.$

##  4 Evaluating the deviation function

This section concerns calculating the deviation function of Theorem 2.1.

######  Theorem 4.1.

Suppose that $F$ is an absolutely continuous function and $I^{Q}(q)<\infty$ .
Then $q$ is absolutely continuous, $\dot{q}\in\mathbb{L}_{2}(\mathbb{R}_{+})$
, the infimum in (2.8) is attained uniquely and

$I^{Q}(q)=\frac{1}{2}\,\int_{0}^{\infty}\overline{p}(t)\bigl{(}\dot{q}(t)-\int_{0}^{t}\dot{q}(s)\,\mathbf{1}_{\\{q(s)>0\\}}\,F^{\prime}(t-s)\,ds+(\beta-
q_{0}^{-})F_{0}^{\prime}(t)\bigr{)}\,dt\,,$

where $\overline{p}(t)$ represents the unique
$\mathbb{L}_{2}(\mathbb{R}_{+})$–solution of the Fredholm equation of the
second kind

$(\mu+\sigma^{2})p(t)=\dot{q}(t)-\int_{0}^{t}\dot{q}(s)\,\mathbf{1}_{\\{q(s)>0\\}}\,F^{\prime}(t-s)\,ds+(\beta-
q_{0}^{-})F_{0}^{\prime}(t)\\\
+\sigma^{2}\int_{0}^{\infty}p(s)F^{\prime}(\lvert
s-t\rvert)\,ds-\sigma^{2}\int_{0}^{\infty}\int_{0}^{s\wedge
t}F^{\prime}(s-r)\,F^{\prime}(t-r)\,dr\,p(s)\,ds\,,$ (4.1)

with $\dot{q}$ , $F_{0}^{\prime}$ and $F^{\prime}$ denoting derivatives.

###### Proof.

On writing

$\int_{\mathbb{R}_{+}^{2}}\,\mathbf{1}_{\\{s+x\leq t\\}}\,\,\dot{k}(\mu
s,F(x))\,dF(x)\,\mu\,ds=\int_{0}^{t}\int_{0}^{F(t-s)}\dot{k}(x,\mu
s)\,dx\,\mu\,ds\,,$ (4.2)

the equation (2.9) is seen to be of the form

$q(t)=f(t)+\int_{0}^{t}q(t-s)^{+}\,dF(s)\,,t\in\mathbb{R}_{+}\,,$

with the functions $f(t)$ and $F(t)$ being absolutely continuous. The function
$q(t)$ is absolutely continuous by Lemma B.1.

It is also noteworthy that (4.2) implies that

$\frac{d}{dt}\,\int_{\mathbb{R}_{+}^{2}}\,\mathbf{1}_{\\{s+x\leq
t\\}}\,\,\dot{k}(\mu
s,F(x))\,dF(x)\,\mu\,ds=\int_{0}^{t}\dot{k}(\mu(t-s),F(s))F^{\prime}(s)\,\mu
ds\,.$ (4.3)

The infimum in (2.8) is attained uniquely by coercitivity and strict convexity
of the function being minimised, cf., Proposition II.1.2 in Ekeland and Temam
[8]. Differentiation in (2.9) with the account of (4.3) obtains that, a.e.,

$\dot{w}^{0}(F_{0}(t))F_{0}^{\prime}(t)+\sigma\,\dot{w}(t)-\int_{0}^{t}F^{\prime}(t-s)\sigma\,\dot{w}(s)\,ds+\int_{0}^{t}\dot{k}(\mu(t-s),F(s))F^{\prime}(s)\,\mu
ds\\\
-\bigl{(}\dot{q}(t)-\int_{0}^{t}\dot{q}(s)\,\mathbf{1}_{\\{q(s)>0\\}}\,F^{\prime}(t-s)\,ds-(q_{0}^{-}-\beta)F_{0}^{\prime}(t)\bigr{)}=0\,.$
(4.4)

Introduce the map

$\Phi:\,(\dot{w}^{0},\dot{w},\dot{k})\to\bigl{(}\dot{w}^{0}(F_{0}(t))F_{0}^{\prime}(t)+\sigma\,\dot{w}(t)-\int_{0}^{t}F^{\prime}(t-s)\sigma\,\dot{w}(s)\,ds+\int_{0}^{t}\dot{k}(\mu(t-s),F(s))F^{\prime}(s)\,\mu
ds\,,t\in\mathbb{R}_{+}\bigr{)}.$

Since $F_{0}^{\prime}(t)$ is bounded by (2.2), $\Phi$ maps
$V=\mathbb{L}_{2}([0,1])\times\mathbb{L}_{2}(\mathbb{R}_{+})\times\mathbb{L}_{2}(\mathbb{R}_{+}\times[0,1])$
to $\mathbb{L}_{2}(\mathbb{R}_{+})$ . For instance, on using that
$\int_{0}^{\infty}F^{\prime}(s)\,ds=1$ ,

$\int_{0}^{\infty}\bigl{(}\int_{0}^{t}F^{\prime}(t-s)\dot{w}(s)\,ds\bigr{)}^{2}\,dt\leq\int_{0}^{\infty}\int_{0}^{t}F^{\prime}(t-s)\dot{w}(s)^{2}\,ds\,dt=\int_{0}^{\infty}\dot{w}(s)^{2}\,ds<\infty$

and

$\int_{0}^{\infty}\bigl{(}\int_{0}^{t}\dot{k}(\mu(t-s),F(s))F^{\prime}(s)\,\mu
ds\bigr{)}^{2}\,dt\leq\int_{0}^{\infty}\int_{0}^{t}\dot{k}(\mu(t-s),F(s))^{2}F^{\prime}(s)\,\mu^{2}ds\,dt\\\
=\mu^{2}\int_{0}^{\infty}\int_{0}^{1}\dot{k}(x,t)^{2}dx\,dt<\infty\,.$

The method of Lagrange multipliers, more specifically, Proposition III.5.2 in
Ekeland and Temam [8] with $Y=\mathbb{L}_{2}(\mathbb{R}_{+})$ and the set of
nonnegative functions as cone $\mathcal{C}$ , yields

$I^{Q}(q)=\sup_{p\in\mathbb{L}_{2}(\mathbb{R}_{+})}\inf_{\begin{subarray}{c}(\dot{w}^{0},\dot{w},\dot{k})\in\mathbb{L}_{2}([0,1])\\\
\times\mathbb{L}_{2}(\mathbb{R}_{+})\times\mathbb{L}_{2}(\mathbb{R}_{+}\times[0,1])\end{subarray}}\Bigl{(}\frac{1}{2}\int_{0}^{1}\dot{w}^{0}(x)^{2}\,dx+\frac{1}{2}\int_{0}^{\infty}\dot{w}(t)^{2}\,dt+\frac{1}{2}\int_{0}^{\infty}\int_{0}^{1}\dot{k}(x,t)^{2}\,dx\,dt\\\
+\int_{0}^{\infty}p(t)\bigl{(}\dot{q}(t)-\int_{0}^{t}\dot{q}(s)\,\mathbf{1}_{\\{q(s)>0\\}}\,F^{\prime}(t-s)\,ds+(\beta-
q_{0}^{-})F_{0}^{\prime}(t)-\dot{w}^{0}(F_{0}(t))F_{0}^{\prime}(t)\\\
-\sigma\,\dot{w}(t)+\int_{0}^{t}F^{\prime}(t-s)\sigma\,\dot{w}(s)\,ds-\int_{0}^{t}\dot{k}(\mu(t-s),F(s))F^{\prime}(s)\,\mu
ds\bigr{)}\bigr{)}\,dt\,.$ (4.5)

Minimising in (4.5) yields

$\displaystyle\overline{\dot{w}^{0}}(x)-\overline{p}(F_{0}^{-1}(x))=0\,,$
(4.6)
$\displaystyle\overline{\dot{w}}(t)-\sigma\overline{p}(t)+\sigma\int_{0}^{\infty}\overline{p}(t+s)F^{\prime}(s)\,ds=0\,,$
(4.7)
$\displaystyle\overline{\dot{k}}(x,t)-\overline{p}(\frac{t}{\mu}+F^{-1}(x))=0\,.$
(4.8)

(For the latter, note that

$\int_{0}^{\infty}p(t)\int_{0}^{t}\dot{k}(\mu(t-s),F(s))F^{\prime}(s)\,\mu
ds\,dt=\int_{0}^{\infty}\int_{s}^{\infty}p(t)\dot{k}(\mu(t-s),F(s))F^{\prime}(s)\,\mu\,dt\,ds\\\
=\int_{0}^{\infty}\int_{0}^{\infty}p(t+s)\dot{k}(\mu
t,F(s))F^{\prime}(s)\,\mu\,dt\,ds=\int_{0}^{\infty}\int_{0}^{1}p(\frac{t}{\mu}+F^{-1}(x))\dot{k}(x,t)\,dx\,dt\,.)$

Hence,

$I^{Q}(q)=\sup_{p\in\mathbb{L}_{2}(\mathbb{R}_{+})}\bigl{(}\int_{0}^{\infty}p(t)\bigl{(}\dot{q}(t)-\int_{0}^{t}\dot{q}(s)\,\mathbf{1}_{\\{q(s)>0\\}}\,F^{\prime}(t-s)\,ds+(\beta-
q_{0}^{-})F_{0}^{\prime}(t)\bigr{)}\,dt\\\
-\frac{1}{2}\bigl{(}\int_{0}^{\infty}p(s)^{2}F_{0}^{\prime}(s)\,ds+\int_{0}^{\infty}(\sigma
p(t)-\sigma\int_{0}^{\infty}p(t+s)F^{\prime}(s)\,ds)^{2}\,dt+\mu\int_{0}^{\infty}\int_{0}^{\infty}p(t+s)^{2}\,F^{\prime}(s)ds\,dt\bigr{)}\bigr{)}\,.$

Noting that

$\int_{0}^{\infty}p(s)^{2}F_{0}^{\prime}(s)\,ds+\mu\int_{0}^{\infty}\int_{0}^{\infty}p(t+s)^{2}\,F^{\prime}(s)ds\,dt=\mu\int_{0}^{\infty}p(s)^{2}\,ds$

yields

$I^{Q}(q)=\sup_{p\in\mathbb{L}_{2}(\mathbb{R}_{+})}\bigl{(}\int_{0}^{\infty}p(t)\bigl{(}\dot{q}(t)-\int_{0}^{t}\dot{q}(s)\,\mathbf{1}_{\\{q(s)>0\\}}\,F^{\prime}(t-s)\,ds+(\beta-
q_{0}^{-})F_{0}^{\prime}(t)\bigr{)}\,dt\\\
-\frac{1}{2}\bigl{(}\mu\int_{0}^{\infty}p(s)^{2}\,ds+\int_{0}^{\infty}(\sigma
p(t)-\sigma\int_{0}^{\infty}p(t+s)F^{\prime}(s)\,ds)^{2}\,dt\bigr{)}\bigr{)}\,.$
(4.9)

The existence and uniqueness of a maximiser in (4.9) follows by Proposition
II.1.2 in Ekeland and Temam [8] because the expression in the $\sup$ tends to
$-\infty$ as $\lVert p\rVert_{\mathbb{L}_{2}(\mathbb{R}_{+})}\to\infty$ .
Varying $p$ in (4.9) yields (4.1). As the maximiser in (4.9) is unique, so is
an $\mathbb{L}_{2}(\mathbb{R}_{+})$–solution of the Fredholm equation.

######  Remark 4.1.

Picard iterations should produce the solution to (4.1).

∎

##  Appendix A Large deviation convergence and idempotent processes

This section reviews the basics of LD convergence and idempotent processes,
see, e.g., Puhalskii [26]. Let $\mathbb{E}$ represent a metric space. Let
$\mathcal{P}(\mathbb{E})$ denote the power set of $\mathbb{E}$. Set function
$\Pi:\,\mathcal{P}(\mathbb{E})\to[0,1]$ is said to be a deviability if
$\Pi(E)=\sup_{y\in E}\Pi(\\{y\\}),\,E\subset\mathbb{E},$ where the function
$\Pi(y)=\Pi(\\{y\\})$ is such that $\sup_{y\in\mathbb{E}}\Pi(y)=1$ and the
sets $\\{y\in\mathbb{E}:\,\Pi(y)\geq\gamma\\}$ are compact for all
$\gamma\in(0,1]$. (One can also refer to $\Pi$ as a maxi–measure or an
idempotent probability.) A deviability is a tight set function in the sense
that $\inf_{K\in\mathcal{K}(\mathbb{E})}\Pi(\mathbb{S}\setminus K)=0$ , where
$\mathcal{K}(\mathbb{E})$ stands for the collection of compact subsets of
$\mathbb{E}$ . If $\Xi$ is a directed set and $F_{\xi}\,,\xi\in\Xi\,,$ is a
net of closed subsets of $\mathbb{E}$ that is nonincreasing with respect to
the partial order on $\Xi$ by inclusion, then
$\Pi(\cap_{\xi\in\Xi}F_{\xi})=\lim_{\xi\in\Xi}\Pi(F_{\xi})$ . A property
pertaining to elements of $\mathbb{E}$ is said to hold $\Pi$–a.e. if the value
of $\Pi$ of the set of elements that do not have this property equals 0 .

Function $f$ from $\mathbb{E}$ to metric space $\mathbb{E}^{\prime}$ is called
an idempotent variable. The idempotent distribution of the idempotent variable
$f$ is defined as the set function $\Pi\circ
f^{-1}(\Gamma)=\Pi(f\in\Gamma),\,\Gamma\subset\mathbb{E}^{\prime}$ . If $f$ is
the canonical idempotent variable defined by $f(y)=y$, then it has $\Pi$ as
the idempotent distribution. The continuous images of deviabilities are
deviabilities, i.e., if $f:\,\mathbb{E}\to\mathbb{E}^{\prime}$ is continuous,
then $\Pi\circ f^{-1}$ defined by $\Pi\circ
f^{-1}(E^{\prime})=\Pi(f^{-1}(E^{\prime}))$ is a deviability on
$\mathbb{E}^{\prime}$ , where $E^{\prime}\subset\mathbb{E}^{\prime}$ .
Furthermore, this property extends to the situation where
$f:\,\mathbb{E}\to\mathbb{E}^{\prime}$ is strictly Luzin, i.e., continuous
when restricted to the set $\\{y\in\mathbb{E}:\,\Pi(y)\geq a\\}$ , for
arbitrary $a\in(0,1]$ . Thus, the idempotent distribution of a strictly Luzin
idempotent variables is a deviability. More generally, $f$ is said to be a
Luzin idempotent variable if the idempotent distribution of $f$ is a
deviability. If $f=(f_{1},f_{2})$ , with $f_{i}$ assuming values in
$\mathbb{E}^{\prime}_{i}$ , then the (marginal) idempotent distribution of
$f_{1}$ is defined by
$\Pi^{f_{1}}(y^{\prime}_{1})=\Pi(f_{1}=y^{\prime}_{1})=\sup_{y:\,f_{1}(y)=y^{\prime}_{1}}\Pi(y)$
. The idempotent variables $f_{1}$ and $f_{2}$ are said to be independent if
${\Pi}(f_{1}=y^{\prime}_{1},\,f_{2}=y^{\prime}_{2})=\Pi(f_{1}=y^{\prime}_{1})\Pi(f_{2}=y^{\prime}_{2})$
for all
$(y^{\prime}_{1},y^{\prime}_{2})\in\mathbb{E}^{\prime}_{1}\times\mathbb{E}^{\prime}_{2}$
, so, the joint distribution is the product of the marginal ones. Independence
of finite collections of idempotent variables is defined similarly.

A sequence $Q_{n}$ of probability measures on the Borel $\sigma$–algebra of
$\mathbb{E}$ is said to large dviation (LD) converge at rate $r_{n}$ to
deviability $\Pi$ if for every bounded continuous non-negative function $f$ on
$\mathbb{E}$

$\lim_{n\to\infty}\left(\int_{\mathbb{E}}f(x)^{n}\,Q_{n}(dx)\right)^{1/r_{n}}=\sup_{x\in\mathbb{E}}f(x)\Pi(x).$

Equivalently, one may require that
$\lim_{n\to\infty}Q_{n}(\Gamma)^{1/r_{n}}=\Pi(\Gamma)$ for every Borel set
$\Gamma$ such that $\Pi$ of the interior of $\Gamma$ and $\Pi$ of the closure
of $\Gamma$ agree. If the $Q_{n}$ LD converge to $\Pi$ , then
$\Pi(y)=\lim_{\delta\to
0}\liminf_{n\to\infty}(Q_{n}(B_{\delta}(y))^{1/r_{n}}=\lim_{\delta\to
0}\limsup_{n\to\infty}(Q_{n}(B_{\delta}(y))^{1/r_{n}}$ , for all
$y\in\mathbb{E}$ , where $B_{\delta}(y)$ represents the open ball of radius
$\delta$ about $y$ . (Closed balls may be used as well.) The sequence $Q_{n}$
is said to be exponentially tight of order $r_{n}$ if
$\inf_{K\in\mathcal{K}(\mathbb{E})}\limsup_{n\to\infty}Q_{n}(\mathbb{E}\setminus
K)^{1/r_{n}}=0$ . If the sequence $Q_{n}$ is exponentially tight of order
$r_{n}$ , then there exists subsequence $Q_{n^{\prime}}$ that LD converges at
rate $r_{n^{\prime}}$ to a deviability. Any such deviability will be referred
to as a Large Deviation (LD) limit point of the $Q_{n}$ . Given
$\tilde{E}\subset\mathbb{E}$ , the sequence $Q_{n}$ is said to be
$\tilde{E}$–exponentially tight if it is exponentially tight and
$\tilde{\Pi}(\mathbb{E}\setminus\tilde{E})=0$ , for any LD limit point of
$Q_{n}$ .

It is immediate that $\Pi$ is a deviability if and only if $I(x)=-\ln\Pi(x)$
is a tight deviation function, i.e., the sets
$\\{x\in\mathbb{E}:\,I(x)\leq\gamma\\}$ are compact for all $\gamma\geq 0$ and
$\inf_{x\in\mathbb{E}}I(x)=0$ , and that the $Q_{n}$ LD converge to $\Pi$ at
rate $r_{n}$ if and only if they obey the LDP for rate $r_{n}$ with deviation
function $I$ , i.e., $\liminf_{n\to\infty}(1/r_{n})\,\ln
Q_{n}(G)\geq-\inf_{x\in G}I(x)$ , for all open sets $G$ , and
$\limsup_{n\to\infty}(1/r_{n})\,\ln Q_{n}(F)\leq-\inf_{x\in F}I(x)$ , for all
closed sets $F$ .

LD convergence of probability measures can be also expressed as LD convergence
in distribution of the associated random variables to idempotent variables.
Sequence $\\{X_{n},\,n\in\mathbb{N}\\}$ of random variables with values in
$\mathbb{E}^{\prime}$ is said to LD converge in distribution at rate $r_{n}$
as $n\to\infty$ to idempotent variable $X$ with values in
$\mathbb{E}^{\prime}$ if the sequence of the probability laws of the $X_{n}$
LD converges to the idempotent distribution of $X$ at rate $r_{n}$. If random
variables $X^{\prime}_{n}$ and $X^{\prime\prime}_{n}$ are independent,
$X^{\prime}_{n}$ LD converges to $X^{\prime}$ and $X^{\prime\prime}_{n}$ LD
converges to $X^{\prime\prime}$ , then the sequence
$(X^{\prime}_{n},X^{\prime\prime}_{n})$ LD converges to
$(X^{\prime},X^{\prime\prime})$ . If sequence $\\{{P}_{n},\,n\in\mathbb{N}\\}$
of probability measures LD converges to deviability ${\Pi}$ , then one has LD
convergence in distribution of the canonical idempotent variables. A
continuous mapping principle holds: if random variables $X_{n}$ LD converge at
rate $r_{n}$ to idempotent variable $X$ and
$f:\,\mathbb{E}^{\prime}\to\mathbb{E}^{\prime\prime}$ is a continuous
function, then the random variables $f(X_{n})$ LD converge at rate $r_{n}$ to
$f(X)$ , where $\mathbb{E}^{\prime\prime}$ is a metric space. The following
version of Slutsky’s theorem holds.

######  Theorem A.1.

Let $Y_{n}$ be random variables with values in $\mathbb{E}^{\prime\prime}$ and
let $a\in\mathbb{E}^{\prime\prime}$ ; if the sequence $X_{n}$ LD converges to
$X$ and $Y_{n}\to a$ super–exponentially in probability for rate $r_{n}$ ,
i.e., $P(d(Y_{n},a)>\epsilon)^{1/r_{n}}\to 0$ , for arbitrary $\epsilon>0$ ,
then the sequence $(X_{n},Y_{n})$ LD converges at rate $r_{n}$ to $(X,a)$ in
$\mathbb{E}^{\prime}\times\mathbb{E}^{\prime\prime}$ , where $d$ denotes the
metric on $\mathbb{E}^{\prime\prime}$ .

Collection $(X_{t},\,t\in\mathbb{R}_{+})$ of idempotent variables on
$\mathbb{E}$ is called an idempotent process. The functions
$(X_{t}(y),\,t\in\mathbb{R}_{+})$ for various $y\in\mathbb{E}$ are called
trajectories (or paths) of $X$. Idempotent processes are said to be
independent if they are independent as idempotent variables with values in the
associated function spaces. The concepts of idempotent processes with
independent and (or) stationary increments mimic those for stochastic
processes.

Since this paper deals with stochastic processes having rightcontinuous
trajectories with lefthand limits, the underlying space $\mathbb{E}$ may be
assumed to be a Skorohod space $\mathbb{D}(\mathbb{R}_{+},\mathbb{R}^{m})$ ,
for suitable $m$ .

Suppose, $X_{n}=(X_{n}(t)\,,t\in\mathbb{R}_{+})$ is a sequence of stochastic
processes which assumes values in a metric space $E^{\prime}$ with metric
$d^{\prime}$ and has rightcontinuous trajectories with lefthand limits. The
sequence $X_{n}$ is said to be exponentially tight of order $r_{n}$ if the
sequence of the distributions of $X_{n}$ as measures on the Skorohod space
$\mathbb{D}(\mathbb{R}_{+},\mathbb{E}^{\prime})$ is exponentially tight of
order $r_{n}$ . It is said to be $\mathbb{C}$–exponentially tight if any LD
limit point is the law of an idempotent process with continuous trajectories,
i.e.,
$\Pi(\mathbb{D}(\mathbb{R}_{+},\mathbb{E}^{\prime})\setminus\mathbb{C}(\mathbb{R}_{+},\mathbb{E}^{\prime}))=0$
, whenever $\Pi$ is an LD limit point of the laws of $X_{n}$ .

The method of finite–dimensional distributions for LD convergence of
stochastic processes is summarised in the next theorem, Puhalskii [23]. The
proof mimics that in weak convergence theory.

######  Theorem A.2.

If, for all tuples $t_{1}<t_{2}<\ldots<t_{l}$ with the $t_{i}$ coming from a
dense subset of $\mathbb{R}_{+}$ , the sequence of vectors
$(X_{n}(t_{1}),\ldots,X_{n}(t_{l}))$ LD converges in $\mathbb{R}^{l}$ at rate
$r_{n}$ to $(X(t_{1}),\ldots,X(t_{l}))$ and the sequence $X_{n}$ is
$\mathbb{C}$–exponentially tight of order $r_{n}$ , then $X$ is a continuous
path idempotent process and the sequence $X_{n}$ LD converges in
$\mathbb{D}(\mathbb{R}_{+},\mathbb{E}^{\prime})$ at rate $r_{n}$ to $X$ .

######  Theorem A.3.

Suppose $\mathbb{E}^{\prime}$ is, in addition, complete and separable. The
sequence $X_{n}$ is $\mathbb{C}$–exponentially tight of order $r_{n}$ if and
only if

  1. 1.

the sequence $X_{n}(t)$ is exponentially tight of order $r_{n}$ , for all $t$
from a dense subset of $\mathbb{R}_{+}$ , and,

  2. 2.

for all $\epsilon>0$ and $L>0$ ,

$\lim_{\delta\to
0}\limsup_{n\to\infty}\sup_{t\in[0,L]}P(\sup_{s\in[0,\delta]}d^{\prime}(X_{n}(t+s),X_{n}(t))>\epsilon)^{1/r_{n}}=0\,.$
(A.1)

###### Proof.

The necessity of the condition follows from the continuity of the projection
mapping and the continuous mapping principle. The proof of sufficiency is
standard and is included for completeness. Let, for $L>0$ , $\delta>0$ and $X$
from the Skorohod space $\mathbb{D}(\mathbb{R}_{+},\mathbb{E}^{\prime})$ ,

$w_{L}(X,\delta)=\sup_{t,s\in[0,L]:\,\lvert
t-s\rvert\leq\delta}d^{\prime}(X(t),X(s))\,.$ (A.2)

Since

$P(w_{L}(X_{n},\delta)>\epsilon)\leq P(\bigcup_{i=1}^{\lfloor
L/\delta\rfloor+1}\\{3\sup_{t\in[i\delta,(i+1)\delta]}d^{\prime}(X_{n}(t),X_{n}(i\delta))>\epsilon\\}\\\
\leq\sum_{i=1}^{\lfloor
L/\delta\rfloor+1}P(3\sup_{t\in[i\delta,(i+1)\delta]}d^{\prime}(X_{n}(t),X_{n}(i\delta))>\epsilon)\leq(\lfloor\frac{L}{\delta}\rfloor+1)\sup_{t\in[0,L]}P(\sup_{s\in[t,t+\delta]}d^{\prime}(X_{n}(s),X_{n}(t))>\frac{\epsilon}{3})\,,$

the hypotheses imply that

$\lim_{\delta\to
0}\limsup_{n\to\infty}P(w_{L}(X_{n},\delta)>\epsilon)^{1/r_{n}}=0\,.$ (A.3)

Let also

$w^{\prime}_{L}(X,\delta)=\inf_{\begin{subarray}{c}0=t_{0}<t_{1}<\ldots<t_{k}=L:\\\
t_{j}-t_{j-1}>\delta\end{subarray}}\,\,\max_{j=1,\ldots,k}\sup_{u,v\in[t_{j-1},t_{j})}d^{\prime}(X(u),X(v))\,.$
(A.4)

Since $w^{\prime}_{L}(X,\delta)\leq w_{L}(X,2\delta)$ , provided $\delta<L/2$
, by (A.3),

$\lim_{\delta\to
0}\limsup_{n\to\infty}P(w_{L}^{\prime}(X_{n},\delta)>\epsilon)^{1/r_{n}}=0\,.$
(A.5)

As each $X_{n}$ is a member of the Skorohod space
$\mathbb{D}(\mathbb{R}_{+},\mathbb{E}^{\prime})$,

$\lim_{\delta\to 0}P(w_{L}^{\prime}(X_{n},\delta)>\epsilon)=0\,,$

so, on recalling (A.5),

$\lim_{\delta\to
0}\sup_{n}P(w^{\prime}_{L}(X_{n},\delta)>\epsilon)^{1/r_{n}}=0\,.$ (A.6)

Let $\\{t_{1},t_{2},\ldots\\}$ represent a dense subset of $\mathbb{R}_{+}$
such that $X_{n}(t_{1}),X_{n}(t_{2}),\ldots$ are exponentially tight of order
$r_{n}$ . Since $\mathbb{E}^{\prime}$ is complete and separable, every
probability measure on $\mathbb{E}^{\prime}$ is tight, so, it may be assumed
that there exist compact subsets $K_{1},K_{2},\ldots$ such that, for all $n$ ,

$P(X_{n}(t_{i})\notin K_{i})^{1/r_{n}}<\frac{1}{2^{i}}\,,i\in\mathbb{N}\,.$
(A.7)

Choose positive $\delta_{1},\delta_{2},\ldots$ such that

$P(w_{i}^{\prime}(X_{n},\delta_{i})>\frac{1}{2^{i}})^{1/r_{n}}<\frac{\epsilon}{2^{i}}\,,i\in\mathbb{N}\,.$
(A.8)

The set

$A=\bigcap_{i\in\mathbb{N}}\\{X:\,w^{\prime}_{i}(X,\delta_{i})\leq\frac{1}{2^{i}}\,,X(t_{i})\in
K_{i}\\}$ (A.9)

has a compact closure, see, e.g., Theorem A2.2 on p. 563 in Kallenberg [16],
and

$P(X_{n}\notin
A)^{1/r_{n}}\leq\sum_{i=1}^{\infty}P(w_{i}(X_{n},\delta_{i})>\frac{1}{2^{i}})^{1/r_{n}}+\sum_{i=1}^{\infty}P(X_{n}(t_{i})\notin
K_{i})^{1/r_{n}}<2\epsilon\,.$ (A.10)

Thus, the sequence $X_{n}$ is exponentially tight of order $r_{n}$ in
$\mathbb{D}(\mathbb{R}_{+},\mathbb{E}^{\prime})$ . Let $\Pi$ represent a
deviability on $\mathbb{D}(\mathbb{R}_{+},\mathbb{E}^{\prime})$ that is an LD
limit point of the distributions of $X_{n}$ . It is proved next that
$\Pi(X)=0$ if $X$ is a discontinuous function. Suppose $X$ has a jump at $t$ ,
i.e., $d^{\prime}(X(t),X(t-))>0$ . Let $\rho$ denote a metric in
$\mathbb{D}(\mathbb{R}_{+},\mathbb{E}^{\prime})$ . It may be assumed that if
$\rho(X^{\prime},X)<\delta$ , then there exists a continuous nondecreasing
function $\lambda(t)$ such that $\sup_{s\leq
t+1}d^{\prime}(X^{\prime}(\lambda(s)),X(s))<\delta$ and $\sup_{s\leq
t+1}\lvert\lambda(s)-s\rvert<\delta$ , see, e.g., Ethier and Kurtz [9], Jacod
and Shiryaev [15]. Then, assuming that $2\delta<t$ and $\delta<1$ ,
$\inf_{s^{\prime}\in[s-\delta,s+\delta]}d^{\prime}(X(s^{\prime}),X(s))<\delta$
, where $s\in[0,t+1]$ . Note that

$d^{\prime}(X(t),X(t-))\leq
d^{\prime}(X(t),X^{\prime}(s_{1}))+d(X^{\prime}(s_{1}),X^{\prime}(t))+d^{\prime}(X^{\prime}(t),X^{\prime}(s_{2}))+d^{\prime}(X^{\prime}(s_{2}),X(s_{3}))\\\
+d^{\prime}(X(s_{3}),X(t-))$

so that, with $s_{1}\in[t-\delta,t+\delta]$ such that
$d^{\prime}(X^{\prime}(s_{1}),X(t))<\delta$ , $s_{3}\in[t-\delta,t]$ such that
$d^{\prime}(X(s_{3}),X(t-))<\delta$ , and
$s_{2}\in[s_{3}-\delta,s_{3}+\delta]$ such that
$d^{\prime}(X^{\prime}(s_{2}),X(s_{3}))<\delta$ ,

$d^{\prime}(X(t),X(t-))<3\delta+2\sup_{s\in[t-2\delta,t+\delta]}d(X^{\prime}(s),X^{\prime}(t))\,,$

which implies that, for $\delta$ small enough,

$\sup_{s\in[t-2\delta,t+\delta]}d^{\prime}(X^{\prime}(s),X^{\prime}(t))>\frac{d^{\prime}(X(t),X(t-))}{3}\,.$
(A.11)

Since

$\sup_{s\in[t-2\delta,t+\delta]}d^{\prime}(X^{\prime}(s),X^{\prime}(t))\leq\sup_{s\in[t,t+\delta]}d^{\prime}(X^{\prime}(s),X^{\prime}(t))+\sup_{s\in[t-2\delta,t-\delta]}d^{\prime}(X^{\prime}(s),X^{\prime}(t))\\\
+\sup_{s\in[t-\delta,t]}d^{\prime}(X^{\prime}(s),X^{\prime}(t))\leq\sup_{s\in[t,t+\delta]}d^{\prime}(X^{\prime}(s),X^{\prime}(t))+\sup_{s\in[t-2\delta,t-\delta]}d^{\prime}(X^{\prime}(s),X^{\prime}(t-2\delta))\\\
+\sup_{s\in[t-\delta,t]}d^{\prime}(X^{\prime}(s),X^{\prime}(t-\delta))+d^{\prime}(X^{\prime}(t-\delta),X^{\prime}(t-2\delta))+2d^{\prime}(X^{\prime}(t),X^{\prime}(t-\delta))\,,$

for $\epsilon=d^{\prime}(X(t),X(t-))/18$ ,

$P(\rho(X_{n},X)<\delta)\leq
P(\sup_{s\in[t,t+\delta]}d^{\prime}(X_{n}(s),X_{n}(t))>\epsilon)+P(\sup_{s\in[t-2\delta,t-\delta]}d^{\prime}(X_{n}(s),X_{n}(t-2\delta))>\epsilon)\\\
+P(\sup_{s\in[t-\delta,t]}d^{\prime}(X_{n}(s),X_{n}(t-\delta))>\epsilon)\,.$

Therefore, assuming $L\geq t$ and $r_{n}\geq 1$ ,

$P(\rho(X_{n},X)<\delta)^{1/r_{n}}\leq
P(\sup_{s\in[t,t+\delta]}d^{\prime}(X_{n}(s),X_{n}(t))>\epsilon)^{1/r_{n}}+P(\sup_{s\in[t-\delta,t]}d^{\prime}(X_{n}(s),X_{n}(t-\delta))>\epsilon)^{1/r_{n}}\\\
+P(\sup_{s\in[t-2\delta,t-\delta]}d^{\prime}(X_{n}(t),X_{n}(t-2\delta))>\epsilon)^{1/r_{n}}\leq
3\sup_{t^{\prime}\in[0,L]}P(\sup_{s\in[0,\delta]}d^{\prime}(X_{n}(t^{\prime}+s),X_{n}(t^{\prime}))>\epsilon)^{1/r_{n}}\,.$

Since

$\Pi(X)=\lim_{\delta\to
0}\liminf_{n\to\infty}P(\rho(X_{n},X)<\delta)^{1/r_{n}}\,,$

(A.1) implies that $\Pi(X)=0$ . ∎

The discussion below concerns the properties of the idempotent processes that
feature prominently in the paper. All the processes assume values in
$\mathbb{R}$ . The standard Wiener idempotent process denoted by
$W=(W(t)\,,t\in\mathbb{R}_{+})$ is defined as an idempotent process with
idempotent distribution

$\Pi^{W}(w)=\exp\bigl{(}-\frac{1}{2}\int_{0}^{\infty}\dot{w}(t)^{2}\,dt\bigr{)}$
(A.12)

provided
$w=(w(t)\,,t\in\mathbb{R}_{+})\in\mathbb{D}(\mathbb{R}_{+},\mathbb{R})$ is
absolutely continuous and $w(0)=0$, and $\Pi^{W}(w)=0$ otherwise. The
restriction to $[0,t]$ produces a standard Wiener idempotent process on
$[0,t]$ which is specified by the idempotent distribution
$\Pi_{t}^{W}(w)=\exp\bigl{(}-1/2\,\int_{0}^{t}\dot{w}(s)^{2}\,ds\bigr{)}$ .
The Brownian bridge idempotent process on $[0,1]$ denoted by
$W^{0}=(W^{0}(x),\,x\in[0,1])$ is defined as an idempotent process with the
idempotent distribution

$\Pi^{W^{0}}(w^{0})=\exp\bigl{(}-\frac{1}{2}\int_{0}^{1}\dot{w}^{0}(x)^{2}\,dx\bigr{)}$
(A.13)

provided $w^{0}=(w^{0}(x)\,,x\in[0,1])\in\mathbb{D}([0,1],\mathbb{R})$ is
absolutely continuous and $w^{0}(0)=w^{0}(1)=0$, and $\Pi^{W^{0}}(w^{0})=0$
otherwise. The Brownian sheet idempotent process on
$[0,1]\times\mathbb{R}_{+}$ denoted by
$(B(x,t)\,,x\in[0,1],\,t\in\mathbb{R}_{+})$ is defined as a two–parameter
idempotent process with the distribution

$\Pi^{B}(b)=\exp\bigl{(}-\frac{1}{2}\int_{[0,1]\times\mathbb{R}_{+}}\dot{b}(x,t)^{2}\,dx\,dt\bigr{)}$
(A.14)

provided $b=(b(x,t)\,,x\in[0,1]\,,t\in\mathbb{R}_{+})$ is absolutely
continuous with respect to Lebesgue measure on $\mathbb{R}_{+}\times[0,1]$ and
$b(x,0)=b(0,t)=0$, and $\Pi^{B}(b)=0$ otherwise. The Kiefer idempotent process
on $[0,1]\times\mathbb{R}_{+}$ denoted by
$(K(x,t)\,,x\in[0,1]\,,t\in\mathbb{R}_{+}),$ is defined as a two–parameter
idempotent process with the idempotent distribution

$\Pi^{K}(k)=\exp\bigl{(}-\frac{1}{2}\int_{[0,1]\times\mathbb{R}_{+}}\dot{k}(x,t)^{2}\,dx\,dt\bigr{)}$
(A.15)

provided $k=(k(x,t)\,,x\in[0,1]\,,t\in\mathbb{R}_{+})$ is absolutely
continuous with respect to Lebesgue measure on $\mathbb{R}_{+}\times[0,1]$ and
$k(0,t)=k(1,t)=k(x,0)=0$, and $\Pi^{K}(k)=0$ otherwise. In this paper we view
it as an element of
$\mathbb{D}(\mathbb{R}_{+},\mathbb{D}([0,1],\mathbb{R}_{+}))$ . Furthermore,
as the deviabilities that $W$ , $W^{0}$ , $B$, or $K$ have discontinuous paths
are equal to zero, these idempotent processes can be considered as having
paths from $\mathbb{C}(\mathbb{R}_{+},\mathbb{R})$ ,
$\mathbb{C}([0,1],\mathbb{R})$ ,
$\mathbb{C}(\mathbb{R}_{+},\mathbb{C}([0,1],\mathbb{R}))$ and
$\mathbb{C}(\mathbb{R}_{+},\mathbb{C}([0,1],\mathbb{R}))$ , respectively.

Being LD limits of their stochastic prototypes, these idempotent processes
have similar properties, as summarised in the next lemma.

######  Lemma A.1.

  1. 1.

For $x>0$ , the idempotent process $(B(x,t)/\sqrt{x}\,,t\in\mathbb{R}_{+})$ is
a standard Wiener idempotent process.

  2. 2.

For $t>0$ , $(K(x,t)/\sqrt{t}\,,x\in[0,1])$ is a Brownian bridge idempotent
process. For $x\in(0,1)$ , the idempotent process
$(K(x,t)/\sqrt{x(1-x)}\,,t\in\mathbb{R}_{+})$ is a standard Wiener idempotent
process.

  3. 3.

The Kiefer idempotent process can be written as

$K(x,t)=-\int_{0}^{x}\frac{K(y,t)}{1-y}\,dy+B(x,t)\,,$ (A.16)

where $B(x,t)$ is a Brownian sheet idempotent process. Conversely, if $B(x,t)$
is a Brownian sheet idempotent process and (A.16) holds, then $K(x,t)$ is a
Kiefer idempotent process. Also,

$W^{0}(x)=-\int_{0}^{x}\frac{W^{0}(y)}{1-y}\,dy+W^{\prime}(x)\,,$ (A.17)

where $W^{\prime}(x)$ is a standard Wiener idempotent process.

###### Proof.

Parts 1 and 2 are elementary. For instance,

$\Pi((\frac{K(x,t)}{\sqrt{t}}\,,x\in[0,1])=(w^{0}(x)\,,x\in[0,1]))\\\
=\sup_{k:\,k(x,t)=\sqrt{t}w^{0}(x)\,,x\in[0,1]}\exp\bigl{(}-\frac{1}{2}\int_{\mathbb{R}_{+}}\int_{0}^{1}\dot{k}(x,s)^{2}\,dx\,ds\bigr{)}\,.$

An application of the Cauchy–Schwarz inequality shows that the optimal $k$ is

$k(x,s)=\frac{s\wedge t}{\sqrt{t}}\,w^{0}(x)\,.$

As for $(K(x,t)/\sqrt{x(1-x)}\,,t\in\mathbb{R}_{+})$ , the optimal trajectory
$(k(y,t)\,,y\in[0,1])$ to get to $w(t)\sqrt{x(1-x)}$ at $x$ is

$k(y,t)=\begin{cases}w(t)\sqrt{x(1-x)}\,\displaystyle\frac{y}{x}\,,&\text{ if
}y\in[0,x]\,,\\\ \\\ w(t)\sqrt{x(1-x)}\,\displaystyle\frac{1-y}{1-x}\,,&\text{
if }y\in[x,1]\,.\end{cases}$

In order to prove part 3, it suffices to show that if

$k(x,t)=-\int_{0}^{x}\frac{k(y,t)}{1-y}\,dy+b(x,t)\,,$ (A.18)

with $k$ and $b$ being absolutely continuous and with $\Pi^{B}(b)>0$ , then
$k(1,t)=0$ and

$\int_{0}^{\infty}\int_{0}^{1}\dot{k}(x,t)^{2}\,dx\,dt=\int_{0}^{\infty}\int_{0}^{1}\dot{b}(x,t)^{2}\,dx\,dt\,.$
(A.19)

Solving (A.18) yields, for $x<1$ ,

$k(x,t)=(1-x)\int_{0}^{x}\frac{b_{y}(y,t)}{1-y}\,dy\,,$ (A.20)

where $b_{y}(y,t)=\int_{0}^{t}\dot{b}(s,y)\,ds$ . By the Cauchy–Schwarz
inequality,

$\lvert\int_{0}^{x}\frac{b_{y}(y,t)}{1-y}\,dy\rvert\leq\sqrt{\int_{0}^{x}\frac{1}{(1-y)^{2}}\,dy}\sqrt{\int_{0}^{1}b_{y}(y,t)^{2}dy}\leq\frac{\sqrt{t}}{\sqrt{1-x}}\,\sqrt{\int_{[0,1]\times\mathbb{R}_{+}}\dot{b}(y,s)^{2}\,dy\,ds}\,.$
(A.21)

We obtain that $k(x,t)\to 0$ as $x\to 1$ provided $\Pi^{B}(b)>0$ .

Let $k_{t}(x,t)=\int_{0}^{x}\dot{k}(y,t)\,dy$ . We show that if $\Pi^{B}(b)>0$
, then, a.e.,

$\frac{1}{1-x}\,k_{t}(x,t)^{2}\to 0\text{ as }x\to 1\,.$ (A.22)

Given arbitrary $a>0$ , by (A.20),

$\frac{k_{t}(x,t)^{2}}{1-x}=(1-x)\Bigl{(}\int_{0}^{x}\frac{\dot{b}(y,t)\,dy}{1-y}\Bigr{)}^{2}\leq
2(1-x)a^{2}\Bigl{(}\int_{0}^{x}\frac{dy}{1-y}\Bigr{)}^{2}\\\
+2(1-x)\int_{0}^{x}\frac{dy}{(1-y)^{2}}\int_{0}^{x}\dot{b}(y,t)^{2}\,\mathbf{1}_{\\{\lvert\dot{b}(y,t)\rvert>a\\}}\,\,dy\leq
2(1-x)\lvert\ln(1-x)\rvert^{2}a^{2}+2\int_{0}^{x}\dot{b}(y,t)^{2}\,\mathbf{1}_{\\{\lvert\dot{b}(y,t)\rvert>a\\}}\,\,dy\,.$

Since $\int_{0}^{1}\dot{b}(y,t)^{2}\,dy<\infty$ a.e., the latter rightmost
side tends to $0$ a.e. , as $x\to 1$ and $a\to\infty$ .

We are now in a position to prove (A.19). Owing to (A.18),

$\int_{0}^{\infty}\int_{0}^{1}\dot{b}(x,t)^{2}\,dx\,dt=\int_{0}^{\infty}\int_{0}^{1}\Bigl{(}\dot{k}(x,t)^{2}+2\dot{k}(x,t)\frac{k_{t}(x,t)}{1-x}+(\frac{k_{t}(x,t)}{1-x})^{2}\Bigr{)}\,dx\,dt\,.$
(A.23)

Integration by parts with the account of (A.22) yields, for almost all $t$ ,

$\int_{0}^{1}\dot{k}(x,t)\frac{k_{t}(x,t)}{1-x}\,dx=-\int_{0}^{1}k_{t}(x,t)\bigl{(}\frac{\dot{k}(x,t)}{1-x}+\frac{k_{t}(x,t)}{(1-x)^{2}})\,dx$

so that

$\int_{0}^{1}\dot{k}(x,t)\frac{k_{t}(x,t)}{1-x}\,dx=-\frac{1}{2}\,\int_{0}^{1}\frac{k_{t}(x,t)^{2}}{(1-x)^{2}}\,dx\,.$

Recalling (A.23) implies (A.19).∎

##  Appendix B A nonlinear renewal equation

This section is concerned with the properties of the equation

$g(t)=f(t)+\int_{0}^{t}g(t-s)^{+}\,dF(s)\,,t\in\mathbb{R}_{+}\,.$ (B.1)

It is assumed that $f(t)$ is a locally bounded measurable function and that
$F(t)$ is a continuous distribution function on $\mathbb{R}_{+}$ with $F(0)<1$
. The existence and uniqueness of an essentially locally bounded solution
$g(t)$ to (B.1) follows from Lemma B.2 in Puhalskii and Reed [28].

######  Lemma B.1.

If the functions $f$ and $F$ are absolutely continuous with respect to
Lebesgue measure and $F(0)=0$ , then the function $g$ is absolutely continuous
too.

###### Proof.

Use Picard iterations. Let $g_{0}(t)=f(t)$ and

$g_{k}(t)=f(t)+\int_{0}^{t}g_{k-1}(t-s)^{+}\,dF(s)\,.$ (B.2)

The functions $g_{k}$ are seen to be continuous. Let $\epsilon>0$ , $T>0$ and
$0\leq t_{0}\leq t_{1}\leq\ldots\leq t_{l}\leq T$ . Since $g_{k}\to g$ locally
uniformly, see Lemma B.1 in Puhalskii and Reed [28], the function $g$ is
continuous and $\sup_{k}\sup_{t\in[0,T]}\lvert g_{k}(t)\rvert\leq M$ , for
some $M>0$ . Note that

$\lvert\int_{0}^{t_{i}}g_{k-1}(t_{i}-s)^{+}\,dF(s)-\int_{0}^{t_{i-1}}g_{k-1}(t_{i-1}-s)^{+}\,dF(s)\rvert\\\
\leq\int_{t_{i-1}}^{t_{i}}\lvert
g_{k-1}(t_{i}-s)\rvert\,dF(s)+\int_{0}^{T}\,\mathbf{1}_{\\{s\leq
t_{i-1}\\}}\,\lvert g_{k-1}(t_{i}-s)-g_{k-1}(t_{i-1}-s)\rvert\,dF(s)\,.$ (B.3)

Let

$\psi(\delta)=\sup_{0\leq t_{0}\leq t_{1}\leq\ldots\leq t_{l}\leq
T}\\{\sum_{i=1}^{l}\lvert f(t_{i})-f(t_{i-1}\rvert+M\sum_{i=1}^{l}\lvert
F(t_{i})-F(t_{i-1})\rvert:\,\sum_{l=1}^{l}(t_{i}-t_{i-1})\leq\delta\\}$ (B.4)

and

$\phi_{k}(\delta)=\sup_{0\leq t_{0}\leq t_{1}\leq\ldots\leq t_{l}\leq
T}\\{\sum_{i=1}^{l}\lvert
g_{k}(t_{i})-g_{k}(t_{i-1})\rvert:\,\sum_{l=1}^{l}(t_{i}-t_{i-1})\leq\delta\\}\,.$
(B.5)

By (B.2) and (B.3), for $k\geq 1$ ,

$\phi_{k}(\delta)\leq\psi(\delta)+\phi_{k-1}(\delta)F(T)\,.$ (B.6)

Let

$\phi(\delta)=\sup_{0\leq t_{0}\leq t_{1}\leq\ldots\leq t_{l}\leq T}\\{\lvert
g(t_{i})-g(t_{i-1})\rvert:\,\sum_{l=1}^{l}(t_{i}-t_{i-1})\leq\delta\\}\,.$
(B.7)

Suppose that $F(t_{0})<1$ . Since $g_{k}\to g$ locally uniformly, as
$k\to\infty$ , $\phi_{k}(\delta)\to\phi(\delta)$ . Letting $k\to\infty$ in
(B.6) implies that $\phi(\delta)\leq\psi(\delta)/(1-F(t_{0}))$ so that
$\phi(\delta)\to 0$ , as $\delta\to 0$ . Hence, $g(t)$ is absolutely
continuous on $[0,t_{0}]$ . Next, as in Puhalskii and Reed [28], write, for
$t\in[0,t_{0}]$ ,

$g(t+t_{0})=f(t+t_{0})+\int_{t}^{t+t_{0}}g(t+t_{0}-s)\,dF(s)+\int_{0}^{t}g(t+t_{0}-s)\,dF(s)\,.$

By what has been proved, the sum of the first two terms on the righthand side
is an absolutely continuous function of $t$ on $[0,t_{0}]$ . The preceding
argument implies that $g(t+t_{0})$ is absolutely continuous in $t$ on
$[0,t_{0}]$ . Iterating the argument proves the absolute continuity of $g(t)$
on $\mathbb{R}_{+}$ . ∎

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