# Large deviations for stochastic fluid networks with Weibullian tails

Mihail Bazhba Chang-Han Rhee Bert Zwart

Department of Quantitative Economics, UvA, Netherlands, Roetersstraat 11,
<EMAIL_ADDRESS>of Industrial Engineering, Northwestern University,
US, IL 60208-3109<EMAIL_ADDRESS>of Stochastics,
CWI, Netherlands, Science Park 123<EMAIL_ADDRESS>and Eindhoven University
of Technology

###### Abstract

We consider a stochastic fluid network where the external input processes are
compound Poisson with heavy-tailed Weibullian jumps. Our results comprise of
large deviations estimates for the buffer content process in the vector-valued
Skorokhod space which is endowed with the product $J_{1}$ topology. To
illustrate our framework, we provide explicit results for a tandem queue. At
the heart of our proof is a recent sample-path large deviations result, and a
novel continuity result for the Skorokhod reflection map in the product
$J_{1}$ topology.

Keywords. fluid networks, large deviations, Skorodhod map, heavy tails.

Mathematics Subject Classification: 60K25, 60F10.

##  1 Introduction

The past 25 years have witnessed a significant research activity on queueing
systems with heavy tails, but the important case of queueing networks has
received less attention. Early papers focused on generalised Jackson networks
(Baccelli et al., (2005)), monotone separable networks (Baccelli and Foss,
(2004)), and max-plus networks (Baccelli et al., (2004)). Recent work on tail
asymptotics of transient cycle times and waiting times for closed tandem
queueing networks can be seen in Kim and Ayhan, (2015). In two joint papers
with Foss, Masakiyo Miyazawa investigated queue lengths in a queueing network
with feedback in Foss and Miyazawa, (2014) and tandem queueing networks in
Foss and Miyazawa, (2018). Compared to standard queueing networks tracking
movements of discrete customers, fluid networks are somewhat more tractable.
In an early paper, it was recognized that tail asymptotics for downstream
nodes could be obtained by analyzing the busy period of upstream nodes, under
certain assumptions (Boxma and Dumas, (1998)). The case of a tandem fluid
queue where the input to the first node is a Lévy process with regularly
varying jump sizes has been investigated in Lieshout and Mandjes, (2008)
exploiting a Laplace transform expression which is available in that case.

More recently, multidimensional asymptotics for the time-dependent buffer
content vector in a fluid queue fed by compound Poisson processes were
investigated in Chen et al., (2019). The framework in Chen et al., (2019)
allows for the analysis of situations in which a large buffer content may be
caused by multiple big jumps in the input process. Such results were
established before for multiple server queues and fluid queues fed by on-off
sources (see, for example, Zwart et al., (2004), Foss and Korshunov, (2012)).
The results on fluid networks in Chen et al., (2019) were derived assuming
regular variation of the jumps in the arrival processes. Work on fluid
networks with light-tailed input is surveyed in Miyazawa, (2011). The goal of
the present paper is to investigate the case where jumps are semi-exponential
(e.g. of Weibull type $\exp\\{-x^{\alpha}\\}$ with $\alpha\in(0,1)$). This
case is somewhat more difficult to analyze, especially in the case where rare
events of interest are caused by multiple big jumps in the input process, as
exhibited in the case of the multiple server queue (Bazhba et al., (2019)).

We focus on a stochastic fluid network comprised of $d$ nodes, with external
inputs modeled as compound Poisson processes with semi-exponential increments.
We are interested in the event that an arbitrary linear combination of the
buffer contents in the network exceeds a large value. We write this functional
as a mapping of the input processes using the well-known multidimensional
Skorokhod reflection map on the positive orthant (see e.g. Whitt, (2002)), and
apply a sample-path large deviations principle for the superposition of
Poisson processes, which has recently been derived in Bazhba et al., (2020).
This sample-path large deviation principle has been established for Poisson
processes with semi-exponential jumps, and holds in the product $J_{1}$
topology. To apply the contraction principle (the analogue of the continuous
mapping argument in a large deviations context), we need to show that the
Skorokhod map has suitable continuity properties. The $J_{1}$ product topology
is not as strong as the standard $J_{1}$ topology on $\mathbb{R}^{d}$, and it
turns out that continuity can only be established for input processes with
nonnegative jumps. However, this result, presented in Theorem 2.1 below, is
sufficient for our proof strategy to work.

The contraction principle leads to an expression of the rate function which we
analyze in detail. Under some generality, we show that each input process
contributes to a large fluid level by at most one big jump, and the
computation of the rate function is reduced to a concave optimization problem
with $2d$ decision variables. We explicitly solve this optimization problem
for the case $d=2$.

The outline of this paper is as follows: Section 2 contains a description of
our model, the topological space in which the input processes are defined, and
an introduction to the reflection map. In Sections 3, 4, and 5 we present our
main results: upper and lower large deviation bounds for the buffer content
process, logarithmic asymptotics for overflow probabilities of the buffer
content process over fixed times, and an explicit analysis of the two-node
tandem network. Section 6 contains technical proofs. We end this paper with an
appendix where we develop several auxiliary large deviations results.

##  2 Model description and preliminary results

###  2.1 The Model

In this section, we describe our model and we present some preliminary results
that are used in our analysis. We consider a single-class open stochastic
fluid network with $d$ nodes. The total amount of external work that arrives
at station $i$ is equal to
$J^{(i)}(t)\triangleq\sum_{j=1}^{N^{(i)}(t)}J^{(i)}_{j},\ t\in[0,T],$ which is
a compound Poisson process with mean $\mu_{i}$. If no exogenous input is
assigned to node $i$, then set $J^{(i)}(\cdot)\equiv 0$, and
$\mu_{i}\triangleq 0$. We define $\mathcal{J}$ as the subset of nodes that
have an exogenous input. For notational convenience, we assume that the
Poisson processes $\\{N^{(i)}(t)\\}_{t\geq 0},i\in\mathcal{J}$ have unit rate.
We pose an assumption on the distribution of the jump sizes $J^{(i)}_{1}$, for
$i\in\mathcal{J}$, making it semi-exponential:

######  Assumption 1.

For each $i\in\mathcal{J}\subseteq\\{1,\ldots,d\\}$,
$\mathbf{P}\left(J^{(i)}_{1}\geq x\right)=e^{-c_{i}L(x)x^{\alpha}}$ where
$\alpha\in(0,1)$, $c_{i}\in(0,\infty),$ and $L$ is a slowly varying function
such that $L(x)/x^{1-\alpha}$ is non-increasing for sufficiently large $x$.

Recall that $L$ is slowly varying if $L(ax)/x\rightarrow 1$ as
$x\rightarrow\infty$ for each $a>0$. At each node $i\in\\{1,\ldots,d\\}$, the
fluid is processed and released at a deterministic rate $r_{i}$. Fractions of
the processed fluid from each node are then routed to other nodes or leave the
network. We characterize the stochastic fluid network by a four-tuple
$(\mathbf{J},\mathbf{r},Q,\mathbf{X}(0))$, where
$\mathbf{J}(\cdot)=\big{(}J^{(1)}(\cdot),\ldots,J^{(d)}(\cdot)\big{)}^{\intercal}$
is the vector of the assigned input processes at each one of the $d$ nodes,
respectively. The vector
$\mathbf{r}\triangleq(r_{1},\ldots,r_{d})^{\intercal}$ is the vector of
deterministic output rates at the $d$ nodes,
$Q\triangleq[q_{ij}]_{i,j\in\\{1,\ldots,d\\}}$ is a $d\times d$ substochastic
routing matrix, and $\mathbf{X}(0)\triangleq(X^{(1)}(0),\ldots,X^{(d)}(0))$ is
a nonnegative random vector of initial contents at the $d$ nodes. If the
buffer at node $i$ and at time $t$ is nonempty, then there is fluid output
from node $i$ at a constant rate $r_{i}$. On the other hand, if the buffer of
node $i$ is empty at time $t$, the output rate equals the minimum of the
combined (i.e., both external and internal) input rates and the output rate
$r_{i}$.

We now provide more details on the stochastic dynamics of our network. A
proportion $q_{i,j}$ of all output from node $i$ is immediately routed to node
$j$, while the remaining proportion $q_{i}\triangleq 1-\sum_{j=1}^{k}q_{ij}$
leaves the network. We assume that $q_{ii}\triangleq 0$, and the routing
matrix $Q$ is substochastic, so that $q_{ij}\geq 0$, and $q_{i}\geq 0$ for all
$i,j$. We also assume that $Q^{n}\to 0$ as $n\to\infty$ which implies that all
input eventually leaves the network. Let $Q^{\intercal}$ be the transpose
matrix of $Q$. Though we focus on time-dependent behavior, we consider the
scenario that the fluid network is stable, ensuring that a high level of fluid
is a rare event. We guarantee the stability of the network by posing the
following assumption, based on Kella, (1996):

######  Assumption 2.

Let $\boldsymbol{\mu}=(\mu_{1},\ldots,\mu_{d})^{\intercal}$, and assume that
$(\mathrm{I}-Q^{\intercal})\mathbf{r}-\boldsymbol{\mu}>0$.

Due to our model specifics, the buffer content at station $i$ is processed at
a constant rate $r_{i}$ from the $i$-th server; and a proportion $q_{ij}$ is
routed from the $i$-th station to the $j$-th server. Let
$\mathcal{Q}=(\mathrm{I}-Q^{\intercal})$. To define the buffer content process
we first define the potential content vector $\mathbf{X}(t)$,
$\mathbf{X}(t)\triangleq\mathbf{X}(0)+\mathbf{J}(t)-\mathcal{Q}rt,t\geq 0$.
Let $\mathbf{Z}^{(i)}(t)$ denote the buffer content of the $i$-th station at
time $t$. We can define the buffer content process by the so-called reflection
map. We first provide an intuitive description of this map. It is defined in
terms of a pair of processes $(\mathbf{Z},\mathbf{Y})$ that solve the
differential equation

$d\mathbf{Z}(t)=d\mathbf{X}(t)+\mathcal{Q}d\mathbf{Y}(t),\ t\geq 0.$ (2.1)

Here, $\mathbf{Y}(\cdot)$ is non-decreasing and ${\mathbf{Y}}_{i}(t)$ only
increases at times where $\mathbf{Z}_{i}(t)=0$ for all $i$ and all $t$.
Consequently, as we assume $\mathbf{Z}(0)=0$, the buffer content is

$\mathbf{Z}(t)=\mathbf{X}(t)+\mathcal{Q}\mathbf{Y}(t),\ t\geq 0.$ (2.2)

We call the map $\mathbf{X}\mapsto(\mathbf{Y},\mathbf{Z})$ the reflection map.
We now provide a more rigorous definition of this map.

###  2.2 The reflection map with discontinuities

We start with the definition of the reflection map. Let $\mathbbm{D}[0,T]$
denote the Skorokhod space: the space of càdlàg functions over $[0,T]$. Denote
with $\mathbbm{D}^{\uparrow}[0,T]$ the subspace of the Skorokhod space
consisting of non-decreasing functions that are non-negative at the origin.
Note that we use the component-wise partial order on $\mathbbm{D}[0,T]$ and
$\mathbb{R}^{k}$. That is, we write
$x_{1}\triangleq(x_{1}^{(1)},\ldots,x_{1}^{(k)})\leq
x_{2}\triangleq(x_{2}^{(1)},\ldots,x_{2}^{(k)})$ in $\mathbb{R}^{k}$ if
$x_{1}^{(i)}\leq x_{2}^{(i)}$ in $\mathbb{R}$ for all $i\in\\{1,\ldots,k\\}$,
and we write $\xi\leq\zeta$ in $\prod_{i=1}^{k}\mathbbm{D}[0,T]$ if
$\xi(t)\leq\zeta(t)$ in $\mathbb{R}^{k}$ for all $t\in[0,T]$.

######  Definition 2.1.

(Definition 14.2.1 of Whitt, (2002)) For any
$\xi\in\prod_{i=1}^{k}\mathbbm{D}[0,T]$ and any reflection matrix
$\mathcal{Q}=(\mathrm{I}-Q^{\intercal})$, let the feasible regulator set be

$\Psi(\xi)\triangleq\left\\{\zeta\in\prod_{i=1}^{k}\mathbbm{D}^{\uparrow}[0,T]:\xi+\mathcal{Q}\zeta\geq
0\right\\},$

and let the reflection map be

$\mathbf{R}\triangleq(\psi,\phi):\prod_{i=1}^{k}\mathbbm{D}[0,T]\to\prod_{i=1}^{k}\mathbbm{D}[0,T]\times\prod_{i=1}^{k}\mathbbm{D}[0,T],$

with regulator component

$\psi(\xi)\triangleq\inf\left\\{\Psi(\xi)\right\\}=\inf\left\\{w\in\prod_{i=1}^{k}\mathbbm{D}[0,T]:w\in\Psi(\xi)\right\\},$

and content component

$\phi(\xi)\triangleq\xi+\mathcal{Q}\psi(\xi).$

The infimum in the definition of $\psi$ may not exist in general. However, in
Theorem 14.2.1 of Whitt, (2002), it is proven that the reflection map is
properly defined with the component-wise order. That is,

$\psi^{(i)}(\xi)(t)=\inf\\{w_{i}(t)\in\mathbb{R}:w\in\Psi(\xi)\\}\ \text{for
all}\ i\in\\{1,\ldots,k\\}\ \text{and}\ t\in[0,T].$

In addition, the regulator set $\Psi(\xi)$ is non-empty and its infimum is
attained in $\Psi(\xi)$. Now, we state some important results regarding the
properties of $(\phi,\psi)$. The following result gives an explicit
representation of the solution of the Skorokhod problem.

######  Result 2.1.

(Theorem 14.2.1, Theorem 14.2.5 and Theorem 14.2.7 of Whitt, (2002)) If
$Y(\cdot)=\psi(\mathbf{X})(\cdot)$ and
$\mathbf{Z}(\cdot)=\phi(\mathbf{X})(\cdot)$, then
$(\mathbf{Y}(\cdot),\mathbf{Z}(\cdot))$ solves the Skorokhod problem
associated with the equation (2.1). The mappings $\psi$ and $\phi$ are
Lipschitz continuous maps w.r.t. the uniform metric.

The next result is a useful property of the Skorokhod map. It allows us to
describe the discontinuities of the reflection map under some mild
assumptions.

######  Result 2.2.

(Lemma 14.3.3, Corollary 14.3.4 and Corollary 14.3.5 of Whitt, (2002))
Consider $\xi\in\prod_{i=1}^{k}\mathbbm{D}[0,T]$. Let $Disc(\psi(\xi))$ and
$Disc(\phi(\xi))$ denote the sets of discontinuity points of $\psi(\xi)$ and
$\phi(\xi)$, respectively. Then it holds that $Disc(\psi(\xi))\cup
Disc(\phi(\xi))=Disc(\xi)$. In addition, if $\xi$ has only positive jumps,
then $\psi(\xi)$ is continuous and

$\phi(\xi)(t)-\phi(\xi)(t-)=\xi(t)-\xi(t-).$

######  Result 2.3.

(Theorem 14.2.6 of Whitt, (2002)) If $\xi\leq\zeta$ in
$\prod_{i=1}^{d}\mathbbm{D}[0,T]$, $T>0$, then $\psi(\xi)\geq\psi(\zeta)$.

###  2.3 Topologies and large deviations

In this section, we introduce our preliminary results on sample-path large
deviations for the input and the content process. For our results we will use
the $J_{1}$ topology. In order to study networks, we need a multidimensional
functional setting. That is, we work on the functional space
$(\prod_{i=1}^{k}\mathbbm{D}[0,T],\prod_{i=1}^{k}\mathcal{T}_{J_{1}})$ which
is a product space equipped with the product $J_{1}$ topology. The product
topology—which we denote with $\prod_{i=1}^{k}\mathcal{T}_{J_{1}}$—is induced
by the product metric $d_{p}$ which in turn is defined in terms of the $J_{1}$
metric on $\mathbbm{D}[0,T]$. More precisely, for
$\xi,\zeta\in\prod_{i=1}^{k}\mathbbm{D}[0,T]$ such that
$\xi=(\xi^{(1)},\ldots,\xi^{(k)})$ and
$\zeta=(\zeta^{(1)},\ldots,\zeta^{(k)})$ we have that
$d_{p}(\xi,\zeta)\triangleq\sum_{i=1}^{k}d_{J_{1}}(\xi^{(i)},\zeta^{(i)}).$
Whenever we make a statement on convergence w.r.t.
$(\prod_{i=1}^{k}\mathbbm{D}[0,T],\prod_{i=1}^{k}\mathcal{T}_{J_{1}})$, we
mean convergence w.r.t. $d_{p}$. Closed and open sets are understood to be
generated by $d_{p}$.

####  2.3.1 Some useful continuous functions

The following two lemmas are elementary. Their proofs are provided in Appendix
A.

######  Lemma 2.2.

For $\kappa\in\mathbb{R}^{d}$, let
$\Upsilon_{\kappa}:\prod_{i=1}^{d}\mathbbm{D}[0,T]\to\prod_{i=1}^{d}\mathbbm{D}[0,T]$
be such that $\Upsilon_{\kappa}(\xi)(t)=\xi(t)-\kappa t$. Then,

  * i)

$\Upsilon_{\kappa}$ is Lipschitz continuous w.r.t. $d_{p}$,

  * ii)

$\Upsilon_{\kappa}$ is a homeomorphism.

######  Lemma 2.3.

The function $\pi:\prod_{i=1}^{d}\mathbbm{D}[0,T]\mapsto\mathbb{R}^{d}$,
$\pi(\xi)=\xi(T)$ is Lipschitz continuous w.r.t. $d_{p}$.

A key step in our approach is to establish the Lipschitz continuity of the
regulator and the buffer content component maps w.r.t. $d_{p}$. This is
executed in Proposition 2.1 and Theorem 2.1 below. Their proofs are deferred
to Section 6. Recall that $\mathbbm{D}^{\uparrow}[0,T]$ is the subspace of the
Skorokhod space containing non-decreasing paths which are non-negative at the
origin. We say that $\xi\in\mathbbm{D}[0,T]$ is a pure jump function if
$\xi=\sum_{i=1}^{\infty}x_{i}\mathbbm{1}_{[u_{i},T]}$ for some $x_{i}$’s and
$u_{i}$’s such that $x_{i}\in\mathbb{R}$ and $u_{i}\in[0,T]$ for each $i$, and
the $u_{i}$’s are all distinct. Let
$\mathbbm{D}^{\uparrow}_{\leqslant\infty}[0,T]$ be the subspace of
$\mathbbm{D}[0,T]$ consisting of non-decreasing pure jump functions that
assume non-negative values at the origin. Subsequently, let
$\mathbbm{D}^{\uparrow}_{\leqslant
k}[0,T]\triangleq\\{\xi\in\mathbbm{D}[0,T]:\xi=\sum_{i=1}^{k}x_{i}\mathbbm{1}_{[u_{i},T]},x_{i}\geq
0,u_{i}\in[0,T],i=1,\ldots,k\\}$ be the subset of
$\mathbbm{D}^{\uparrow}_{\leqslant\infty}[0,T]$ containing pure jump functions
of at most $k$ jumps. In addition, for $\beta\in\mathbb{R}$, let
$\mathbbm{D}^{\beta}_{\leqslant
k}[0,T]\triangleq\\{\zeta\in\mathbbm{D}[0,T]:\zeta(t)=\xi(t)+\beta\cdot t,\
\xi\in\mathbbm{D}^{\uparrow}_{\leqslant k}[0,T]\\}$ and
$\mathbbm{D}^{\beta}_{\leqslant\infty}[0,T]\triangleq\\{\zeta\in\mathbbm{D}[0,T]:\zeta(t)=\xi(t)+\beta\cdot
t,\ \xi\in\mathbbm{D}^{\uparrow}_{\leqslant\infty}[0,T]\\}.$ Let
$\mathbbm{D}_{\leqslant k}[0,T]$ denote the subspace of $\mathbbm{D}[0,T]$
consisting of paths with at most $k$ jumps, i.e. $\mathbbm{D}_{\leqslant
k}[0,T]=\\{\xi\in\mathbbm{D}[0,T]:|Disc(\xi)|\leq k\\}$. Finally, let
$\mathbbm{D}^{\beta}[0,T]\triangleq\\{\zeta\in\mathbbm{D}[0,T]:\zeta(t)=\xi(t)+\beta\cdot
t,\ \xi\in\mathbbm{D}^{\uparrow}[0,T]\\}$.

######  Proposition 2.1.

Let $\beta_{i}\in\mathbb{R}$ for $i=1,\ldots,d$. The (restriction of the)
regulator map
$\psi:\prod_{i=1}^{d}\mathbbm{D}^{\beta_{i}}[0,T]\to\prod_{i=1}^{d}\mathbbm{D}[0,T]$
is Lipschitz continuous w.r.t. $d_{p}$.

######  Theorem 2.1.

Let $\beta_{i}\in\mathbb{R}$ for $i=1,\ldots,d$. The (restriction of the)
reflection map
$\mathbf{R}:\prod_{i=1}^{d}\mathbbm{D}^{\beta_{i}}[0,T]\to\prod_{i=1}^{2d}\mathbbm{D}[0,T]$,
where $\mathbf{R}=(\phi,\psi)$, is Lipschitz continuous w.r.t. $d_{p}$.

Note that the restriction of the domain to the paths without downward jumps is
essential for this type of results to hold. Since the order in which the jumps
take place matters for the action of the reflection map, we cannot ensure the
continuity of the reflection map without such extra regularity conditions. The
main difficulty arises with paths which have jumps with different signs in
multiple coordinates appearing almost simultaneously (K. Ramanan, personal
communication).

####  2.3.2 The extended sample-path LDP for the content process

Let $(\mathbb{S},d)$ be a metric space, and $\mathcal{T}$ denote the topology
induced by the metric $d$. Let $\\{X_{n}\\}$ be a sequence of
$\mathbb{S}$-valued random variables. Let $I$ be a non-negative lower semi-
continuous function on $\mathbb{S}$, and $\\{a_{n}\\}$ be a sequence of
positive real numbers that tends to infinity as $n\to\infty$.

######  Definition 2.4.

The probability measures of $(X_{n})$ satisfy an _extended_ LDP in
$(\mathbb{S},d)$ with speed $a_{n}$ and rate function $I$ if

$-\inf_{x\in
A^{\circ}}I(x)\leq\liminf_{n\to\infty}\frac{\log\mathbf{P}(X_{n}\in
A)}{a_{n}}\leq\limsup_{n\to\infty}\frac{\log\mathbf{P}(X_{n}\in
A)}{a_{n}}\leq-\lim_{\epsilon\to 0}\inf_{x\in A^{\epsilon}}I(x)$

for any measurable set $A$.

Here we denote
$A^{\epsilon}\triangleq\\{\xi\in\mathbb{S}:d(\xi,A)\leq\epsilon\\}$ where
$d(\xi,A)=\inf_{\zeta\in A}d(\xi,\zeta)$. The notion of an extended LDP has
been introduced in Borovkov and Mogulskii, (2010) and is quite useful in the
setting of semi-exponential random variables, in which a full LDP is provably
impossible, as is shown in Bazhba et al., (2020).

We finish this section by stating an extended large deviations principle for
the multidimensional input process of the stochastic fluid network. Recall
that $\mathbf{J}$ denotes the input process which is a vector of independent
compound Poisson processes with mean vector $\boldsymbol{\mu}$. We consider
the scaled version
$\bar{\mathbf{J}}_{n}(\cdot)\triangleq\frac{1}{n}\mathbf{J}(n\cdot)$. For any
$\xi\in\prod_{i=1}^{d}\mathbbm{D}[0,T]$, let

$I(\xi^{(j)})=\sum_{\\{t:\xi^{(j)}(t)\neq\xi^{(j)}(t-)\\}}\left(\xi^{(j)}(t)-\xi^{(j)}(t-)\right)^{\alpha}.$

The next result is an immediate consequence of Theorem 2.3 and Remark 2.2 in
Bazhba et al., (2020), combined with Lemma B.1.

######  Result 2.4.

The probability measures of $\bar{\mathbf{J}}_{n}$ satisfy the extended LDP in
$\Big{(}\prod_{i=1}^{d}\mathbbm{D}^{\uparrow}[0,T],\prod_{i=1}^{d}\mathcal{T}_{J_{1}}\Big{)}$
with speed $L(n)n^{\alpha}$, and with rate function
${I}^{(d)}:\prod_{i=1}^{d}\mathbbm{D}^{\uparrow}[0,T]\to[0,\infty]$, where

${I}^{(d)}(\xi^{(1)},...,\xi^{(d)})=\begin{cases}\sum_{j\in\mathcal{J}}c_{j}I(\xi^{(j)})&\text{if
}\xi^{(j)}\in\mathbbm{D}^{\mu_{j}}_{\leqslant\infty}[0,T]\text{ for}\
j\in\mathcal{J}\\\ &\text{and}\ \xi^{(j)}\equiv 0\ \text{for}\
j\notin\mathcal{J},\\\ \infty&otherwise.\\\ \end{cases}$ (2.3)

Recall that the potential content vector is a function of the exogenous input
plus the internal input; that is,
$\mathbf{X}(t)=\mathbf{J}(t)-\mathcal{Q}\mathbf{r}t$. We define the scaled
version of the potential content vector
$\mathbf{X}_{n}(\cdot)=\frac{1}{n}\mathbf{X}(n\cdot)$. Obviously,
$\mathbf{X}_{n}$ is the image of $\mathbf{J}_{n}$ where the map
$\Upsilon_{\mathcal{Q}\mathbf{r}}$ is applied. Due to Lemma 2.2,
$\Upsilon_{\mathcal{Q}\mathbf{r}}$ is Lipschitz continuous and is a
homeomorphism with respect to the product $J_{1}$ metric. The following large
deviation principle for $\mathbf{X}_{n}(\cdot)$ is a direct consequence of
Result 2.4 and ii) of Lemma B.3.

######  Result 2.5.

The probability measures of $\mathbf{X}_{n}$ satisfy an extended LDP in
$\left(\prod_{i=1}^{d}\mathbbm{D}^{-(\mathcal{Q}\mathbf{r})_{i}}[0,T],\prod_{i=1}^{d}\mathcal{T}_{J_{1}}\right)$
with speed $L(n)n^{\alpha}$ and with rate function

$\tilde{I}^{(d)}(\xi^{(1)},...,\xi^{(d)})=\begin{cases}\sum_{j\in\mathcal{J}}c_{j}I(\xi^{(j)})&\text{if
}\xi^{(j)}\in\mathbbm{D}^{(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{i}}_{\leqslant\infty}[0,T]\
\text{for}\ j\in\mathcal{J}\\\ &\text{and}\
\xi^{(j)}(t)=-(\mathcal{Q}\mathbf{r})_{j}(t)\ \text{for}\
j\notin\mathcal{J},\\\ \infty&otherwise.\\\ \end{cases}$ (2.4)

In the next sections, we state our main results along with some additional
useful lemmas.

##  3 Large deviations for the buffer content process

In this section, we state large deviation bounds for the scaled buffer content
process $\mathbf{Z}_{n}(\cdot)\triangleq\frac{1}{n}\mathbf{Z}(n\cdot)$ with
$\mathbf{Z}$ defined in (2.2). The reflection map enables us to represent the
buffer content process in terms of the potential content process
$\mathbf{X}_{n}(\cdot)\triangleq\frac{1}{n}\mathbf{X}(n\cdot)$ and the map
$\phi$ by $\mathbf{Z}_{n}=\phi(\mathbf{X}_{n})$. We apply an analogue of the
contraction principle for extended LDP’s (Lemma B.3) to obtain asymptotic
estimates for the probability measures of $(\mathbf{Z}_{n})$:

######  Theorem 3.1.

The probability measures of $\mathbf{Z}_{n}$ satisfy:

  * i)

For any set $F$ that is closed in
$\left(\prod_{i=1}^{d}\mathbbm{D}[0,T],\prod_{i=1}^{d}\mathcal{T}_{J_{1}}\right)$,

$\limsup_{n\to\infty}\frac{1}{L(n)n^{\alpha}}\log\mathbf{P}\left(\mathbf{Z}_{n}\in
F\right)\leq-\lim_{\epsilon\to 0}\inf_{\xi\in F^{\epsilon}}I_{S}(\xi).$

  * ii)

For set $G$ that is open in
$\left(\prod_{i=1}^{d}\mathbbm{D}[0,T],\prod_{i=1}^{d}\mathcal{T}_{J_{1}}\right)$,

$\liminf_{n\to\infty}\frac{1}{L(n)n^{\alpha}}\log\mathbf{P}\left(\mathbf{Z}_{n}\in
G\right)\geq-\inf_{\xi\in G}I_{S}(\xi),$

where

$I_{S}(\zeta)=\inf\left\\{\tilde{I}^{(d)}(\xi):\zeta=\phi(\xi),\
\xi\in\prod_{i=1}^{d}\mathbbm{D}^{-(\mathcal{Q}\mathbf{r})_{i}}[0,T]\right\\}.$

Note that $I_{S}$ may not be lower semi-continuous, because $\tilde{I}$ is not
a good rate function; see Bazhba et al., (2020) for details.

###### Proof.

Theorem 2.1 ensures that $\phi$ is Lipschitz continuous w.r.t. $d_{p}$. The
bounds therefore follows immediately from Lemma B.3 and Result 2.5. ∎

The function $I_{S}$ is the solution of a constrained minimization problem
over step functions, with a concave objective function, and a constraint that
depends on the solution of the Skorokhod problem displayed in Theorem 3.1.
Though this Skorokhod problem only needs to be evaluated for step functions,
this minimization problem is in general not tractable. To get more concrete
results we look at more specific functionals of the buffer content process in
subsequent sections.

##  4 Asymptotics for overflow probabilities

This section examines the probability that the buffer content associated with
a subset of nodes in the system exceeds a high level. In particular, we fix
$\mathbf{b}=(b_{1},\ldots,b_{d})\in\mathbb{R}_{+}^{d}$ and consider a
functional $\mathscr{B}:\mathbbm{D}[0,T]\to\mathbb{R}_{+}$ defined as
$\mathscr{B}(\eta)\triangleq\mathbf{b}^{\intercal}\pi(\phi(\eta))=\mathbf{b}^{\intercal}\phi(\eta)(T)$
for $\eta\in\mathbbm{D}[0,T]$. We study the linear combination of the buffer
content at fixed time given by
$\mathbf{b}^{\intercal}\mathbf{Z}_{n}(T)=\mathscr{B}(\mathbf{X}_{n})$. Let

$I^{\prime}(x)\triangleq\inf\left\\{\tilde{I}^{(d)}(\xi_{1},\ldots,\xi_{d}):\mathscr{B}(\xi)=x,\
\xi\in\prod_{i=1}^{d}\mathbbm{D}^{-(\mathcal{Q}\mathbf{r})_{i}}[0,T]\right\\},$

define the set
$V_{\geqslant}(y)\triangleq\\{\xi\in\prod_{i=1}^{d}\mathbbm{D}^{(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{i}}_{\leqslant
1}[0,T]:\mathscr{B}(\xi)\geq y\\},$ and let $V_{\geqslant}^{*}(y)$ be the
optimal value of $\tilde{I}^{(d)}$ over the set $V_{\geqslant}(y)$, i.e.
$V_{\geqslant}^{*}(y)\triangleq\inf_{\xi\in
V_{\geqslant}(y)}\tilde{I}^{(d)}(\xi)$. Similarly, let
$V_{>}(y)\triangleq\\{\xi\in\prod_{i=1}^{d}\mathbbm{D}^{(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{i}}_{\leqslant
1}[0,T]:\mathscr{B}(\xi)>y\\}$ and set $V_{>}^{*}(y)\triangleq\inf_{\xi\in
V_{>}(y)}\tilde{I}^{(d)}(\xi).$ Note that $V^{*}_{\geqslant}(y)$ and
$V^{*}_{>}(y)$ depend on $T$, but we suppress the dependence for notational
simplicity.

The next lemma, which is proven in Section 6, enables us to reduce the
feasible region of the optimization problem $\inf_{x\geq y}I^{\prime}(x)$ from
$\prod_{i=1}^{d}\mathbbm{D}^{(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{i}}_{\leqslant\infty}[0,T]$
to
$\prod_{i=1}^{d}\mathbbm{D}^{(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{i}}_{\leqslant
1}[0,T]$, i.e. we can restrict our class of functions to those that have at
most one discontinuity in each coordinate.

######  Lemma 4.1.

Let
$\xi\in\prod_{i=1}^{d}\mathbbm{D}^{(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{i}}_{{\leqslant\infty}}[0,T]$.
Then, there exists a path
$\tilde{\xi}\in\prod_{i=1}^{d}\mathbbm{D}^{(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{i}}_{\leqslant
1}[0,T]$ such that

  * i)

$\tilde{I}^{(d)}(\tilde{\xi})\leq\tilde{I}^{(d)}(\xi)$,

  * ii)

$\phi(\tilde{\xi})(T)\geq\phi(\xi)(T)$,

Since $\tilde{I}^{(d)}(\xi)=\infty$ for
$\xi\notin\prod_{i=1}^{d}\mathbbm{D}^{(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{i}}_{{\leqslant\infty}}[0,T]$,
an immediate consequence of Lemma 4.1 is that

$V_{\geqslant}^{*}(y)=\inf_{x\in[y,\infty)}I^{\prime}(x)\qquad\text{and}\qquad
V_{>}^{*}(y)=\inf_{x\in(y,\infty)}I^{\prime}(x).$ (4.1)

Recall that $\mathcal{J}$ is the set of nodes with exogenous input. Next, let
$I^{+}=\\{j\in\\{1,\ldots,d\\}:b_{j}>0\\}$. The following lemmas, proven in
Section 6, ensure the continuity of $\mathrm{V}^{*}_{\geqslant}(\cdot)$.

######  Lemma 4.2.

Assume that $\mathcal{J}\cap I^{+}\neq\emptyset$. The map $x\mapsto
V_{\geqslant}^{*}(x)$ is $\alpha$-Hölder continuous:

$|V^{*}_{\geqslant}(y)-V^{*}_{\geqslant}(x)|\leq\max_{\\{1\leq i\leq
d:b_{i}>0\\}}\frac{c_{i}}{b_{i}^{\alpha}}\cdot|y-x|^{\alpha}.$

######  Lemma 4.3.

Assume that $\mathcal{J}\cap I^{+}\neq\emptyset$. It holds that
$V^{*}_{\geqslant}(y)=V^{*}_{>}(y)$.

For a given $y>0$, let $P_{\mathbf{b},y}^{*}$ denote the optimal value of the
following optimization problem $\eqref{Pb}$:

$\displaystyle\inf_{(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}}$
$\displaystyle\sum_{i=1}^{d}c_{i}x_{i}^{\alpha}$ $\displaystyle s.t.$
$\displaystyle\mathscr{B}\left(x_{1}\mathbbm{1}_{[u_{1},T]}(\cdot)+(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{1}(\cdot),\
\ldots,\
x_{d}\mathbbm{1}_{[u_{d},T]}(\cdot)+(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{d}(\cdot)\right)\geq
y,$ $\displaystyle x_{i}\geq 0\ \text{for}\ i\in\mathcal{J},\ x_{i}=0\
\text{for}\ i\notin\mathcal{J},$ $\displaystyle u_{i}\in[0,T]\ for\
i=1,\ldots,d.$

For a general stochastic network with routing matrix $Q$ and reflection matrix
$\mathcal{Q}=(\mathrm{I}-Q^{\intercal})$, the following logarithmic
asymptotics hold:

######  Theorem 4.1.

For a fixed $\mathbf{b}=(b_{1},\ldots,b_{d})\in\mathbb{R}_{+}^{d}$ assume that
$\mathcal{J}\cap I^{+}\neq\emptyset$. The overflow probabilities
$\mathbf{P}\left(\mathscr{B}(\mathbf{Z}_{n})\geq y\right)$ satisfy the
following logarithmic asymptotics:

$\lim_{n\to\infty}\frac{1}{L(n)n^{\alpha}}\log\mathbf{P}(\mathscr{B}(\mathbf{Z}_{n})\geq
y)=-V_{\geqslant}^{*}(y)$ (4.2)

Moreover, $V_{\geqslant}^{*}(y)$ is equal to the optimal value
$P_{\mathbf{b},y}^{*}$ of the optimization problem $\eqref{Pb}$ above.

###### Proof.

Note first that $\mathscr{B}$ is Lipschitz continuous w.r.t. $d_{p}$, since it
is a linear functional of the composition of $\pi$ and $\phi$, which are
Lipschitz continuous (from Lemma 2.3 and Theorem 2.1). This allows us to
proceed with applying Lemma B.3 i) to deduce asymptotic upper and lower bounds
for $\frac{1}{L(n)n^{\alpha}}\log\mathbf{P}(\mathscr{B}(\mathbf{Z}_{n})\geq
y)$.

For the upper bound, recall that
$I^{\prime}(y)=\inf\\{\tilde{I}^{(d)}(\xi):\mathscr{B}(\xi)=y\\}$. Thanks to
Result 2.5, the upper bound in Lemma B.3 i), and (4.1), we have that

$\limsup_{n\to\infty}\frac{1}{L(n)n^{\alpha}}\log\mathbf{P}(\mathscr{B}(\mathbf{Z}_{n})\geq
y)\leq-\lim_{\epsilon\to
0}\inf_{x\in[y-\epsilon,\infty)}I^{\prime}(x)=-\lim_{\epsilon\to
0}V^{*}_{\geqslant}\left(y-\epsilon\right)=-V^{*}_{\geqslant}\left(y\right),$

where the last equality is from the continuity of $V^{*}_{\geqslant}(\cdot)$
in Lemma 4.2.

For the lower bound, we apply Result 2.5, the lower bound in Lemma B.3 i),
(4.1), and Lemma 4.3 to obtain

$\displaystyle\liminf_{n\to\infty}\frac{1}{L(n)n^{\alpha}}\log\mathbf{P}(\mathscr{B}(\mathbf{Z}_{n})\geq
y)$
$\displaystyle\geq\liminf_{n\to\infty}\frac{1}{L(n)n^{\alpha}}\log\mathbf{P}(\mathscr{B}(\mathbf{Z}_{n})>y)$
$\displaystyle\geq-\inf_{x\in(y,\infty)}I^{\prime}(x)=-V_{>}^{*}(y)=-V_{\geqslant}^{*}(y).$

To conclude that $V_{\geqslant}^{*}(y)=P_{\mathbf{b},y}^{*}$, we apply Lemma
4.1 to show that their feasible regions are identical. We observe that for
$\xi(\cdot)=\big{(}x_{1}\mathbbm{1}_{[u_{1},T]}(\cdot)+(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{1}(\cdot),\
\ldots,\
x_{d}\mathbbm{1}_{[u_{d},T]}(\cdot)+(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{d}(\cdot)\big{)}$,
we obtain $\tilde{I}^{(d)}(\xi)=\sum_{i=1}^{d}c_{i}x_{i}^{\alpha}$, which
equals the cost function of $(P_{y})$. ∎

Note that large deviations behavior is a consequence of at most one jump in
the external input process to each node. Consequently, the rate function is
the solution of a $2d$-dimensional optimization problem where one has to
optimize the size as well as the time of the jumps. This provides a
significant reduction in complexity compared to the setting of Section 3.
Nevertheless, (4) is still rather intricate in general: it is an
$L^{\alpha}$-norm minimization problem with $\alpha\in(0,1)$. In general, such
problems are strongly NP-hard, cf. Ge et al., (2011). In addition, checking
whether a solution to (4) is feasible requires one to compute values of the
Skorokhod map for step functions, which is nontrivial. To get more explicit
results and gain some physical insights, we therefore consider a two-node
tandem network in the next section.

##  5 A two-node example

We consider a two-node tandem network where content from node 1 flows into
node 2, and content from node 2 leaves the system, i.e. $q_{12}=1$, and
$q_{ij}=0$ otherwise. We assume that each node has an exogenous input process
(i.e. $\mathcal{J}=\\{1,2\\}$). We consider the problem of estimating
$\mathbf{P}\left(\mathbf{Z}^{(2)}_{n}(T)\geq y\right).$ Theorem 4.1 implies
that this is equivalent to solving (4) with $\mathbf{b}=(0,1)$, which is the
subject of this section. To keep the presentation concise, we give an outline
of the key steps and focus on physical insight.

We first develop an explicit expression for the buffer content at time $T$ for
input processes of the form $\xi_{i}(t)=\mu_{i}t+x_{i}\mathbbm{1}(u_{i}\geq
t),t\in[0,T]$, $x_{i}\geq 0$, $u_{i}\in[0,T],i=1,2$. To develop physical
intuition is it instructive to write the buffer content process at node 2 as
the solution of a one-dimensional reflection mapping, fed by the superposition
of $\xi_{2}$ and the output process of node $1$, which in turn is governed by
a one-dimensional reflection mapping as well. To this end, observe that
$z_{1}(t)=\xi_{1}(t)-\inf_{s\leq t}\xi_{1}(s)$. The amount of fluid flowing
from node $1$ to node $2$ in $[0,t]$ is given by
$\xi_{1}(t)-z_{1}(t)=-\inf_{s\leq t}\xi_{1}(s)$. Consequently, we can write

$z_{2}(T)=\xi_{2}(T)-\inf_{s\leq T}\xi_{1}(s)-\inf_{u\leq
T}(\xi_{2}(u)-\inf_{s\leq u}\xi_{1}(s)).$ (5.1)

Our goal is to minimize the cost $c_{1}x_{1}^{\alpha}+c_{2}x_{2}^{\alpha}$
subject to the constraint $z_{2}(T)\geq y$, over $x_{1}\geq 0,\ x_{2}\geq 0,\
u_{1}\in[0,T],\ u_{2}\in[0,T]$. We simplify this problem by identifying
convenient choices of $u_{1}$ and $u_{2}$ which do not lose optimality.

To this end, observe that a jump of size $x_{2}$ at time $u_{2}$ can instead
take place at time $u_{2}=T$ without decreasing $z_{2}(T)$. To determine a
convenient choice of $u_{1}$, note that a jump of size $x_{1}$ in node 1 at
time $u_{1}$ causes an outflow of rate $r_{1}$ from node $1$ to node $2$ in
the interval $[u_{1},u_{1}+x_{1}/(r_{1}-\mu_{1})]$, and rate $\mu_{1}$ after
time $u_{1}+x_{1}/(r_{1}-\mu_{1})$. Therefore, we can take $u_{1}$ such that
$u_{1}+x_{1}/(r_{1}-\mu_{1})=T$, without decreasing $z_{2}(T)$. This choice is
feasible as long as $u_{1}$ remains non-negative, i.e. we require that
$x_{1}/(r_{1}-\mu_{1})\leq T$. Observe that choosing $x_{1}/(r_{1}-\mu_{1})>T$
would not be optimal, as it would increase the cost term involving
$x_{1}^{\alpha}$ without increasing $z_{2}(T)$.

We proceed by solving (5.1) by taking
$\xi_{1}(t)=\mu_{1}t+x_{1}\mathbbm{1}(t\geq T-x_{1}/(r_{1}-\mu_{1}))$ and
$\xi_{2}(t)=\mu_{2}t+x_{2}\mathbbm{1}(t=T)$. Straightforward manipulations
show that

$z_{2}(T)=x_{2}+(r_{1}+\mu_{2}-r_{2})^{+}\frac{x_{1}}{r_{1}-\mu_{1}}.$ (5.2)

We see that a jump at node 1 has no effect on the buffer content in node 2 if
$r_{2}\geq r_{1}+\mu_{1}$, which is intuitively obvious since node 2 is still
rate stable when the output of node $1$ equals $r_{1}$. Therefore, our first
conclusion is that

$\lim_{n\to\infty}\frac{1}{L(n)n^{\alpha}}{\log\mathbf{P}\left(\mathbf{Z}^{(2)}_{n}(T)\geq
y\right)}=-c_{2}y^{\alpha},\hskip 28.45274ptr_{2}\geq r_{1}+\mu_{2}.$ (5.3)

We now turn to the more interesting case $r_{2}<r_{1}+\mu_{2}$. We do not lose
optimality if the constraint on $z_{2}(T)$ is tight, so we can impose the
constraints

$x_{2}+\frac{r_{1}+\mu_{2}-r_{2}}{r_{1}-\mu_{1}}x_{1}=y,\hskip
14.22636ptx_{1}\in[0,(r_{1}-\mu_{1})T],\hskip 14.22636ptx_{2}\geq 0.$ (5.4)

From convex optimization theory, see Corollary 32.3.2 in Rockafellar, (1970),
the minimum of the concave objective function
$c_{1}x_{1}^{\alpha}+c_{2}x_{2}^{\alpha}$ subject to the constraints (5.4) is
achieved over the extreme points of (5.4). In our particular situation, this
implies that an optimal solution should correspond to one of the following 3
cases: (i) $x_{1}=0$, (ii) $x_{2}=0$, (iii) $x_{1}=(r_{1}-\mu_{1})T$. In case
$(iii)$ we would have $x_{2}=y-(r_{1}+\mu_{2}-r_{2})T$, which is only feasible
if $y\geq(r_{1}+\mu_{2}-r_{2})T$.

If $y\leq(r_{1}+\mu_{2}-r_{2})T$, we can conclude that either case (i) holds
with $x_{1}=0,x_{2}=y$, and cost $c_{2}y^{\alpha}$, or case (ii) holds with
$x_{2}=0$, $x_{1}=y\frac{r_{1}-\mu_{1}}{r_{1}+\mu_{2}-r_{2}}$, and cost
$c_{1}\left(y\frac{r_{1}-\mu_{1}}{r_{1}+\mu_{2}-r_{2}}\right)^{\alpha}$. We
conclude that

$\lim_{n\to\infty}\frac{1}{L(n)n^{\alpha}}{\log\mathbf{P}\left(\mathbf{Z}^{(2)}_{n}(T)\geq
y\right)}=-\min\left\\{c_{1}\left(\frac{r_{1}-\mu_{1}}{r_{1}+\mu_{2}-r_{2}}\right)^{\alpha},c_{2}\right\\}y^{\alpha},\hskip
5.69046pty\leq(r_{1}+\mu_{2}-r_{2})T.$ (5.5)

We now turn to the case $y>(r_{1}+\mu_{2}-r_{2})T$. In this case, the time
horizon $T$ is small w.r.t. $y$: the output of node $1$ alone is never enough
to cause the buffer content of node $2$ to reach level $y$ at time $T$. Thus,
we must compare case (i) and case (iii). Case (i) has solution $x_{2}=y$ with
cost $c_{2}y^{\alpha}$. Case (iii) has solution $x_{1}=(r_{1}-\mu_{1})T$,
$x_{2}=y-(r_{1}+\mu_{2}-r_{2})T$, with cost
$c_{1}((r_{1}-\mu_{1})T)^{\alpha}+c_{2}(y-(r_{1}+\mu_{2}-r_{2})T)^{\alpha}$.
We conclude that, if $y>(r_{1}+\mu_{2}-r_{2})T$,

$\lim_{n\to\infty}\frac{1}{L(n)n^{\alpha}}{\log\mathbf{P}\left(\mathbf{Z}^{(2)}_{n}(T)\geq
y\right)}=-\min\\{c_{2}y^{\alpha},c_{1}((r_{1}-\mu_{1})T)^{\alpha}+c_{2}(y-(r_{1}+\mu_{2}-r_{2})T)^{\alpha}\\}.$
(5.6)

To give a numerical example, take $y=2,T=1,r_{1}=r_{2}=3,\mu_{1}=\mu_{2}=1$.
In this case, the inequality $y>(r_{1}+\mu_{2}-r_{2})T$ holds. To evaluate
(5.6), note that the cost of case (i) equals $c_{2}2^{\alpha}$ and the cost of
case (iii) equals $c_{1}2^{\alpha}+c_{2}$. So we conclude that case (iii) is
the most likely way for the event $\\{\mathbf{Z}^{(2)}_{n}(1)\geq 2\\}$ to
occur if $c_{1}\leq c_{2}(1-2^{-\alpha})$, corresponding to a most likely
behavior of two big jumps: $x_{1}=2$, occuring at node 1 at time 0, and
$x_{2}=1$, occuring at node 2 at time 1.

##  6 Complementary proofs

###  6.1 Proofs of Lemma 4.1, 4.2, and 4.3

###### Proof of Lemma 4.1.

Recall that $\mathcal{Q}$ is the reflection matrix, which is invertible with

$\mathcal{Q}^{-1}=(\mathrm{I}-Q^{\intercal})^{-1}=\mathrm{I}+Q+Q^{2}+\ldots.$

Consequently, $\mathcal{Q}^{-1}$ is a matrix with non-negative entries. Next,
we continue with an observation, which we call (O1): if
$u,v\in\mathbb{R}^{k}$, and $A$ is such that $A\in\mathbb{R}_{+}^{k\times k}$,
then $u\geq v$ implies $Au\geq Av$. Let
$\xi=(\xi^{(1)},\ldots,\xi^{(d)})\in\prod_{i=1}^{d}\mathbbm{D}^{(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{i}}_{{\leqslant\infty}}[0,T]$.
Since
$\xi^{(i)}\in\mathbbm{D}^{(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{i}}_{{\leqslant\infty}}[0,T]$,
we have the following representation:

$\xi^{(i)}(t)=(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{i}\cdot
t+\sum_{j=1}^{\infty}x_{j}^{(i)}\mathbbm{1}_{\left[u_{j}^{(i)},T\right]}(t),\
t\in[0,T].$

This representation plays a key role in our proof, which consists of the
following steps.

Step 1: There exists a path
$\tilde{\xi}\in\prod_{i=1}^{d}\mathbbm{D}^{(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{i}}_{\leqslant
1}[0,T]$ such that $\tilde{\xi}\leq\xi$, and $\tilde{\xi}(T)=\xi(T)$. To see
this, let $s^{(i)}=\sup\\{t:\xi^{(i)}(t)-\xi^{(i)}(t-)>0\\}$ for each
$i=1,\ldots,d$, and let

$\tilde{\xi}^{(i)}(t)=(\mu-\mathcal{Q}r)_{i}\cdot
t+\bigg{(}\sum_{j=1}^{\infty}x_{j}^{(i)}\bigg{)}\mathbbm{1}_{\left[s^{(i)},T\right]}(t),\
t\in[0,T].$

By construction, $\tilde{\xi}\leq\xi$ component-wise, and
$\tilde{\xi}(T)=\xi(T)$.

Step 2: It holds that $\psi(\tilde{\xi})(T)\geq\psi(\xi)(T)$, which follows
immediately from Result 2.3.

Step 3: Since $\mathcal{Q}$ consists of non-negative entries, it holds that
$\mathcal{Q}\psi(\tilde{\xi})(T)\geq\mathcal{Q}\psi(\xi)(T)$.

Now, we conclude the proof of our statement. Recall that
$\phi(\xi)(T)=\xi(T)+\mathcal{Q}\psi(\xi)(T)$. From Step 3 and that
$\tilde{\xi}(T)=\xi(T)$ in Step 1, ii) of our lemma is obvious. For i), due to
the sub-additivity of the function $x\mapsto x^{\alpha}$ for $\alpha\in(0,1)$,
we notice that

$\displaystyle\tilde{I}^{(d)}(\tilde{\xi})$
$\displaystyle=\sum_{i=1}^{d}c_{i}\cdot\left[\sum_{\\{t:{\tilde{\xi}}^{(i)}(t)-\tilde{\xi}^{(i)}(t-)>0\\}}(\tilde{\xi}^{(i)}(t)-\tilde{\xi}^{(i)}(t-))^{\alpha}\right]$
$\displaystyle=\sum_{i=1}^{d}c_{i}\cdot\bigg{(}\sum_{j=1}^{\infty}x_{j}^{(i)}\bigg{)}^{\alpha}\leq\sum_{i=1}^{d}c_{i}\cdot\sum_{j=1}^{\infty}\bigg{(}x_{j}^{(i)}\bigg{)}^{\alpha}$
$\displaystyle=\sum_{i=1}^{d}c_{i}\cdot\left[\sum_{\\{t:{\xi}^{(i)}(t)-{\xi}^{(i)}(t-)>0\\}}({\xi}^{(i)}(t)-{\xi}^{(i)}(t-))^{\alpha}\right]=\tilde{I}^{(d)}(\xi).$

∎

Next, we focus on the continuity of $\mathrm{V}^{*}_{\geqslant}(\cdot)$. Let
$\mathbbm{D}_{+}[0,T]$ be the subspace of $\mathbbm{D}[0,T]$ that contains
paths with only positive discontinuities:
$\mathbbm{D}_{+}[0,T]=\\{\xi\in\mathbbm{D}[0,T]:\xi(t)-\xi(t-)\geq 0,\ \forall
t\in[0,T]\\}$. Recall that $\mathbbm{D}_{\leqslant
k}[0,T]=\\{\xi\in\mathbbm{D}[0,T]:|Disc(\xi)|\leq k\\}$.

######  Lemma 6.1.

If $a=(a_{1},\ldots,a_{d})\in\mathbb{R}^{d}_{+}$ and
$\xi\in\prod_{i=1}^{d}(\mathbbm{D}_{+}[0,T]\cap\mathbbm{D}_{\leqslant
1}[0,T])$, let $\zeta=\xi+a\mathbbm{1}_{\\{T\\}}$. Then

  * i)

$\psi(\zeta)=\psi(\xi)$,

  * ii)

$\phi(\zeta)(T)=\phi(\xi)(T)+a$, and

  * iii)

$\tilde{I}^{(d)}(\zeta)\leq\tilde{I}^{(d)}(\xi)+\sum_{i=1}^{d}c_{i}a_{i}^{\alpha}.$

###### Proof.

For i), from the proof of the Theorem 14.2.2 in Whitt, (2002), we see that for
any $\iota\in\prod_{i=1}^{k}\mathbbm{D}[0,T]$ the regulator component
$\psi(\iota)$ is the limit (w.r.t. the uniform metric) of
$\rho_{\iota}^{n}(\mathbf{0})$ where $\mathbf{0}$ is the zero function and
$\rho_{\iota}^{n}$ is the $n$ fold composition of
$\rho_{\iota}:\prod_{i=1}^{d}\mathbbm{D}^{\uparrow}[0,T]\to\prod_{i=1}^{d}\mathbbm{D}^{\uparrow}[0,T]$
such that
$\rho_{\iota}(\eta)(t)=0\vee\sup_{s\in[0,t]}\\{Q\eta(s)-\iota(s)\\}$. We first
claim that $\psi(\zeta)(t)=\psi(\xi)(t)$ for $t\in[0,T-\epsilon]$ for any
fixed $\epsilon>0$. Obviously,
$\rho^{1}_{\zeta}(\mathbf{0})(t)=0\vee\sup_{s\in[0,t]}\\{-\zeta(s)\\},0\\}=0\vee\sup_{s\in[0,t]}\\{-\xi(s)\\},0\\}=\rho^{1}_{\xi}(\mathbf{0})(t)$
for $t\leq T-\epsilon$. To proceed with induction, suppose that
$\rho_{\zeta}^{k}(\mathbf{0})(t)=\rho_{\xi}^{k}(\mathbf{0})(t)$ for
$t\in[0,T-\epsilon]$. Then, for any $t\in[0,T-\epsilon]$ we have

$\rho^{k+1}_{\zeta}(\mathbf{0})(t)=0\vee\sup_{s\in[0,t]}\left\\{Q\rho^{k}_{\zeta}(\mathbf{0})(s)-\zeta(s)\right\\}=0\vee\sup_{s\in[0,t]}\left\\{Q\rho^{k}_{\xi}(\mathbf{0})(s)-\xi(s)\right\\}=\rho^{k+1}_{\xi}(\mathbf{0})(t).$

Since the equality holds for every $k\in\mathbb{N}$, their (uniform) limits
$\psi(\zeta)$ and $\psi(\xi)$ should also coincide on $[0,T-\epsilon]$,
proving the claim. Since $\epsilon$ was arbitrary, this means that
$\psi(\zeta)(t)=\psi(\xi)(t)$, $t\in[0,T)$. Furthermore, since $\psi(\zeta)$
and $\psi(\xi)$ are continuous from Result 2.2, we conclude that they coincide
on $[0,T]$, i.e., $\psi(\xi)=\psi(\zeta)$.

For ii), observe that
$\phi(\zeta)(T)=\zeta(T)+\psi(\zeta)(T)=\xi(T)+a+\psi(\xi)(T)=\phi(\xi)(T)+a.$

For iii), we assume that
$\xi^{(j)}\in\mathbbm{D}^{(\boldsymbol{\mu}-\mathcal{Q}\mathbf{r})_{i}}_{\leqslant\infty}[0,T]$
for $j\in\mathcal{J}$ and $\xi^{(j)}(t)=-(\mathcal{Q}\mathbf{r})_{j}(t)\
\text{for}\ j\notin\mathcal{J}$ since if not
$\tilde{I}^{(d)}(\zeta)=\tilde{I}^{(d)}(\xi)=\infty$, and the inequality holds
trivially otherwise. Let $\zeta=(\zeta^{(1)},\ldots,\zeta^{(d)})$, and
$\xi=(\xi^{(1)},\ldots,\xi^{(d)})$. Recall that the function $x\mapsto
x^{\alpha},\ \alpha\in(0,1)$, is sub-additive, and

$I(\zeta^{(i)})=\sum_{t\in[0,T]:\zeta^{(i)}(t)\neq\zeta^{(i)}(t-)}\left(\zeta^{(i)}(t)-\zeta^{(i)}(t-)\right)^{\alpha}.$

For any $i\in\\{1,\ldots,d\\}$, we have

$\displaystyle I(\zeta^{(i)})$
$\displaystyle=\sum_{t\in[0,T):\xi^{(i)}(t)\neq\xi^{(i)}(t-)}\left(\xi^{(i)}(t)-\xi^{(i)}(t-)\right)^{\alpha}+\left(\xi^{(i)}(T)-\xi^{(i)}(T-)+a_{i}\right)^{\alpha}$
$\displaystyle\leq\sum_{t\in[0,T):\xi^{(i)}(t)\neq\xi^{(i)}(t-)}\left(\xi^{(i)}(t)-\xi^{(i)}(t-)\right)^{\alpha}+\left(\xi^{(i)}(T)-\xi^{(i)}(T-)\right)^{\alpha}+a_{i}^{\alpha}$
$\displaystyle=I(\xi^{(i)})+a_{i}^{\alpha}.$

Therefore,
$\tilde{I}^{(d)}(\zeta)=\sum_{j\in\mathcal{J}}c_{j}I(\zeta^{(j)})\leq\sum_{j\in\mathcal{J}}c_{j}I(\xi^{(j)})+\sum_{j\in\mathcal{J}}c_{j}a_{j}^{\alpha}\leq\tilde{I}^{(d)}(\xi)+\sum_{j=1}^{d}c_{j}a_{j}^{\alpha}.$
∎

###### Proof of Lemma 4.2.

Let $y\geq x\geq 0$. It is obvious that $V^{*}_{\geqslant}(y)\geq
V^{*}_{\geqslant}(x)\geq 0.$ By the definition of the infimum, for any
$\epsilon>0$, there exists a $\zeta\in V_{\geqslant}(x)$ so that
$\tilde{I}^{(d)}(\zeta)<V^{*}_{\geqslant}(x)+\epsilon$. Next, let $i\in I^{+}$
where $I^{+}=\\{j\in\\{1,\ldots,d\\}:b_{j}>0\\}$. Subsequently, let
$\xi=\zeta+\mathbf{v}\mathbbm{1}_{\\{T\\}}$ where
$\mathbf{v}=(0,\ldots,\frac{y-x}{b_{i}},\ldots,0)$. Due to ii) of Lemma 6.1,

$\mathbf{b}^{\intercal}\phi(\xi)(T)=\mathbf{b}^{\intercal}(\phi(\zeta)(T)+\mathbf{v})\geq\mathbf{b}^{\intercal}\phi(\zeta)(T)+b_{i}\frac{(y-x)}{b_{i}}\geq
x+y-x=y.$

Hence, $\xi\in V_{\geqslant}(y)$. Due to iii) of Lemma 6.1,
$\tilde{I}^{(d)}(\xi)\leq\tilde{I}^{(d)}(\zeta)+\max_{\\{1\leq i\leq
d:b_{i}>0\\}}\frac{c_{i}}{b_{i}^{\alpha}}(y-x)^{\alpha}.$ We see that

$V^{*}_{\geqslant}(y)\leq\tilde{I}^{(d)}(\xi)\leq\tilde{I}^{(d)}(\zeta)+\max_{1\leq
i\leq
d:b_{i}>0}\frac{c_{i}}{b_{i}^{\alpha}}(y-x)^{\alpha}<V^{*}_{\geqslant}(x)+\max_{\\{1\leq
i\leq d:b_{i}>0\\}}\frac{c_{i}}{b_{i}^{\alpha}}(y-x)^{\alpha}+\epsilon.$

This leads to $V^{*}_{\geqslant}(y)-V^{*}_{\geqslant}(x)\leq\max_{\\{1\leq
i\leq d:b_{i}>0\\}}\frac{c_{i}}{b_{i}^{\alpha}}(y-x)^{\alpha}+\epsilon$. We
obtain the desired result by letting $\epsilon$ tend to $0$. Thus,
$|V^{*}_{\geqslant}(y)-V^{*}_{\geqslant}(x)|\leq\max_{\\{1\leq i\leq
d:b_{i}>0\\}}\frac{c_{i}}{b_{i}^{\alpha}}\cdot|y-x|^{\alpha}.$ ∎

We end this section with the proof of $V^{*}_{\geqslant}(y)=V^{*}_{>}(y)$.

###### Proof of Lemma 4.3.

For any $\epsilon>0$, we have that $V^{*}_{\geqslant}(y+\epsilon)\geq
V^{*}_{>}(y)$. Hence, in view of Lemma 4.2,

$|V^{*}_{>}(y)-V^{*}_{\geqslant}(y)|=V^{*}_{>}(y)-V^{*}_{\geqslant}(y)\leq
V^{*}_{\geqslant}(y+\epsilon)-V^{*}_{\geqslant}(y)\leq\max_{\\{1\leq i\leq
d:b_{i}>0\\}}\frac{c_{i}}{b_{i}^{\alpha}}|\epsilon|^{\alpha}.$

Now, we let $\epsilon$ go to $0$ to obtain the desired result. ∎

###  6.2 Proof of Theorem 2.1

Recall that $\prod_{i=1}^{d}\mathbbm{D}[0,T]$ is the Skorokhod space equipped
with the product $J_{1}$ topology and
$\mathbbm{D}^{\uparrow}[0,T]\triangleq\\{\xi\in\mathbbm{D}[0,T]:\xi\ \text{is
non-decreasing on $[0,T]$ and $\xi(0)\geq 0$}\\}.$
$\mathbbm{D}^{\uparrow}[0,T]$ is a closed subspace of $\mathbbm{D}[0,T]$
w.r.t. the $J_{1}$ topology. Hence,
$\prod_{i=1}^{d}\mathbbm{D}^{\uparrow}[0,T]$ is a closed subspace of
$\prod_{i=1}^{d}\mathbbm{D}[0,T]$ w.r.t. the product $J_{1}$ topology. Since
$\mathbbm{D}^{\beta}[0,T]$ is the image of $\mathbbm{D}^{\uparrow}[0,T]$ under
the homeomorphism $\Upsilon_{\beta}$, we have that
$\prod_{i=1}^{d}\mathbbm{D}^{\beta_{i}}[0,T]$ is a closed subset of
$\prod_{i=1}^{d}\mathbbm{D}[0,T]$. Let
$\boldsymbol{\beta}=(\beta_{1},\ldots,\beta_{d})$, and let
$\|\boldsymbol{\beta}\|_{\infty}=\max_{1\leq i\leq d}|\beta_{i}|$.

####  6.2.1 Some supporting lemmas

######  Lemma 6.2.

Suppose that $\lambda,\kappa$ are strictly increasing functions such that

  * i)

$\lambda(0)=\kappa(0)=0$,

  * ii)

$\lambda(T)=\kappa(T)=T$,

  * iii)

$\|\lambda-e\|_{\infty}<\delta$, and $\|\kappa-e\|_{\infty}<\delta$.

Then, $\|\lambda\circ\kappa-e\|_{\infty}<2\delta$.

###### Proof.

$\|\lambda\circ\kappa-e\|_{\infty}=\|\lambda-\kappa^{-1}\|_{\infty}\leq\|\lambda-e\|_{\infty}+\|\kappa^{-1}-e\|_{\infty}=\|\lambda-e\|_{\infty}+\|e-\kappa\|_{\infty}\leq
2\delta$. ∎

We now consider properties of continuous and increasing time deformations
$w^{(i)},\ i=1,\ldots,d$.

######  Lemma 6.3.

If $w^{(i)}$ is increasing and continuous so that $w^{(i)}(0)=0$, and
$w^{(i)}(T)=T$ for each $i=1,\ldots,d$, then
$\check{w}(s)=\min\\{w^{(1)}(s),\ldots,w^{(d)}(s)\\}$, and
$\hat{w}(s)=\max\\{w^{(1)}(s),\ldots,w^{(d)}(s)\\}$ are increasing, continuous
such that $\check{w}(0)=\hat{w}(0)=0$, and $\check{w}(T)=\hat{w}(T)=T$.

###### Proof.

The $\min$ and $\max$ of continuous and increasing functions is increasing and
continuous. The other properties are easily verified. ∎

Recall that $\psi$ is Lipschitz continuous w.r.t. uniform norm (Theorem 14.2.5
of Whitt, (2002)). Let $K$ denote the Lipschitz constant of $\psi$ w.r.t. the
uniform norm.

######  Lemma 6.4.

Let $\zeta\in\prod_{i=1}^{d}\mathbbm{D}^{\beta_{i}}[0,T]$ and $w$ be an
increasing function such that $w(0)=0,\ w(T)=T$. Then, it holds that

$\|\psi(\zeta)(w)-\psi(\zeta)(e)\|_{\infty}<K\|\boldsymbol{\beta}\|_{\infty}\|w-e\|_{\infty}.$

###### Proof.

The proof is a consequence of the following two claims:

  * i)

for every $s\in[0,T]$ there exists a $U(s)\in(0,\infty)$ so that
$|\psi(\zeta)(w(s))-\psi(\zeta)(s)|\leq U(s)$, and

  * ii)

it holds that $\sup_{s\in[0,T]}U(s)\leq
K\|\boldsymbol{\beta}\|_{\infty}\|w-\epsilon\|_{\infty}.$

For claim i), pick an $s\in[0,T]$. If $w(s)\geq s$, since $\psi(\zeta)$ is an
increasing function, $\psi(\zeta)(w(s))\geq\psi(\zeta)(s).$ Hence, we only
need to bound $\psi(\zeta)(w(s))-\psi(\zeta)(s)$. Moreover, since
$\zeta\in\prod_{i=1}^{d}\mathbbm{D}^{\beta_{i}}[0,T]$, we have that

$\zeta(w(s))=\zeta((w(s)-s)+s)\geq\zeta(s)+\boldsymbol{\beta}(w(s)-s).$ (6.1)

Next, consider the path $\tilde{\zeta}_{1}$ where

$\tilde{\zeta}_{1}=\begin{cases}\zeta(t),&\ t\in[0,s],\\\
\zeta(s)+\boldsymbol{\beta}(t-s),&\ t\in[s,w(s)].\end{cases}$

Since $\tilde{\zeta}_{1}\leq\zeta$ over $[0,w(s)]$ and Result 2.3, we have
$\psi(\tilde{\zeta}_{1})(w(s))\geq\psi(\zeta)(w(s))$. Furthermore, let

$\tilde{\zeta}_{2}=\begin{cases}\zeta(t),&\ t\in[0,s],\\\ \zeta(s),&\
t\in[s,w(s)].\end{cases}$

By the definition of $\tilde{\zeta}_{2}$ we have that
$\psi(\tilde{\zeta}_{2})(w(s))=\psi(\zeta)(s)$. Moreover, due to the
construction of $\tilde{\zeta}_{1}$,

$\psi(\zeta)(w(s))-\psi(\zeta)(s)\leq\psi(\tilde{\zeta}_{1})(w(s))-\psi(\tilde{\zeta}_{2})(w(s))\leq
K\sup_{t\in[0,w(s)]}|\tilde{\zeta}_{1}(s)-\tilde{\zeta}_{2}(s)|\leq
K\|\boldsymbol{\beta}\|_{\infty}|w(s)-s|.$

Conversely, if $w(s)\leq s$, since $\psi(\zeta)$ is an increasing function,
$\psi(\zeta)(s)\geq\psi(\zeta)(w(s)).$ Furthermore, since
$\zeta\in\prod_{i=1}^{d}\mathbbm{D}^{\beta_{i}}[0,T]$, we have that

$\zeta(s)=\zeta((s-w(s))+w(s))\geq\zeta(w(s))+\boldsymbol{\beta}(s-w(s)).$
(6.2)

Next, consider the path $\tilde{\zeta}_{1}$, where

$\tilde{\zeta}_{1}=\begin{cases}\zeta(t),&\ t\in[0,w(s)],\\\
\zeta(s)+\boldsymbol{\beta}(t-w(s)),&\ t\in[w(s),s].\end{cases}$

Since $\tilde{\zeta}_{1}\leq\zeta$ over $[0,s]$, due to Result 2.3, we have
that $\psi(\tilde{\zeta}_{1})(s)\geq\psi(\zeta)(s)$. On the other hand, let

$\tilde{\zeta}_{2}(t)=\begin{cases}\zeta(t),&\ t\in[0,w(s)],\\\ \zeta(s),&\
t\in[w(s),s].\end{cases}$

Due to the construction of $\tilde{\zeta}_{2}$, we have that
$\psi(\tilde{\zeta}_{2})(w(s))=\psi(\zeta)(s)$. Moreover,

$\psi(\zeta)(w(s))-\psi(\zeta)(s)\leq\psi(\tilde{\zeta}_{1})(w(s))-\psi(\tilde{\zeta}_{2})(w(s))\leq
K\sup_{t\in[0,w(s)]}|\tilde{\zeta}_{1}(s)-\tilde{\zeta}_{2}(s)|\leq
K\|\boldsymbol{\beta}\|_{\infty}|w(s)-s|.$

For claim ii), observe that for every $s\in[0,T]$,
$K\|\boldsymbol{\beta}\|_{\infty}|w(s)-s|\leq
K\|\boldsymbol{\beta}\|_{\infty}\|w-e\|_{\infty}.$ Hence,
$\|\psi(\zeta)(w)-\psi(\zeta)\|_{\infty}\leq
K\|\boldsymbol{\beta}\|_{\infty}\|w-e\|_{\infty}.$ ∎

Note that, if $\boldsymbol{\beta}=\mathbf{0}$ and
$\zeta\in\prod_{i=1}^{d}\mathbbm{D}^{\beta_{i}}[0,T]$, then $\zeta$ belongs to
$\prod_{i=1}^{d}\mathbbm{D}^{\uparrow}[0,T]$ and is non-negative at the
origin. This implies $\psi(\zeta)=0$ and the upper bound in Lemma 6.4 holds
trivially. Next, we state two more lemmas which are needed in our proof for
the Lipschitz continuity of the regulator map in
$\prod_{i=1}^{d}\mathbbm{D}^{\beta_{i}}[0,T].$

######  Lemma 6.5.

Consider $w=(w^{(1)},\ldots,w^{(d)})$, each component of which is a time
deformation. Recall $\hat{w}$ and $\check{w}$ in Lemma 6.3. That is,
$\hat{w}(t)=\max\\{w^{(1)}(t),\ldots,w^{(d)}(t)\\}$, and
$\check{w}(t)=\min\\{w^{(1)}(t),\ldots,w^{(d)}(t)\\}$. For any
$\xi\in\prod_{i=1}^{d}\mathbbm{D}^{\beta_{i}}[0,T]$,

  * i)

$\psi(\xi(w^{(1)}),\ldots,\xi(w^{(d)}))\leq\psi(\xi)(\check{w})+2K\|\boldsymbol{\beta}\|_{\infty}(\|w-e\|_{\infty})$,
and

  * ii)

$\psi(\xi(w^{(1)}),\ldots,\xi(w^{(d)}))+2K\|\boldsymbol{\beta}\|_{\infty}(\|w-e\|_{\infty})\geq\psi(\xi)(\hat{w})$.

###### Proof.

We start with i). Since $\xi\in\prod_{i=1}^{d}\mathbbm{D}^{\beta_{i}}[0,T]$
and $\check{w}(s)\leq\min\\{w^{(1)}(s),\ldots,w^{(d)}(s)\\}$, we have that for
each $i=1,\ldots,d$,

$\xi^{(i)}(w^{(i)}(s))\geq\xi^{(i)}(\check{w}(s))-\|\boldsymbol{\beta}\|_{\infty}(w^{(i)}(s)-\check{w}(s)),\
s\in[0,T].$

Therefore, due to Result 2.3, and the Lipschitz continuity of $\psi$ w.r.t.
the uniform metric,

$\displaystyle\psi(\xi^{(1)}(w^{(1)}),\ldots,\xi^{(d)}(w^{(d)}))$
$\displaystyle\leq\psi(\xi^{(1)}(\check{w})-\|\boldsymbol{\beta}\|_{\infty}(w^{(1)}-\check{w}),\ldots,\xi^{(d)}(\check{w})-\|\boldsymbol{\beta}\|_{\infty}(w^{(d)}-\check{w}))$
$\displaystyle\leq\psi(\xi)(\check{w})+K\|\boldsymbol{\beta}\|_{\infty}\max_{1\leq
i\leq d}(\|\check{w}-w^{(i)}\|_{\infty})$
$\displaystyle\leq\psi(\xi)(\check{w})+2K\|\boldsymbol{\beta}\|_{\infty}\max_{1\leq
i\leq d}(\|w^{(i)}-e\|_{\infty})$
$\displaystyle=\psi(\xi)(\check{w})+2K\|\boldsymbol{\beta}\|_{\infty}(\|w-e\|_{\infty}).$

For ii), observe that
$\xi^{(i)}(\hat{w}(s))\geq\xi^{(i)}({w}^{(i)}(s))-\|\boldsymbol{\beta}\|_{\infty}(\hat{w}(s)-w^{(i)}(s))$
for each $i=1,\ldots,d$ and $s\in[0,T]$, since
$\xi\in\prod_{i=1}^{d}\mathbbm{D}^{\beta_{i}}[0,T]$, and
$\hat{w}(s)\geq\max\\{w^{(1)}(s),\ldots,w^{(d)}(s)\\}$. Therefore, due to
Result 2.3, and the Lipschitz continuity of $\psi$ w.r.t. the uniform metric,

$\displaystyle\psi(\xi^{(1)},\ldots,\xi^{(d)})(\hat{w})$
$\displaystyle\leq\psi(\xi^{(1)}(\hat{w})-\|\boldsymbol{\beta}\|_{\infty}(\hat{w}-w^{(1)}),\ldots,\xi^{(d)}(\hat{w})-\|\boldsymbol{\beta}\|_{\infty}(\hat{w}-w^{(d)}))$
$\displaystyle\leq\psi(\xi)(\hat{w})+K\|\boldsymbol{\beta}\|_{\infty}\max_{1\leq
i\leq d}(\|\hat{w}-w^{(i)}\|_{\infty})$
$\displaystyle\leq\psi(\xi)(\hat{w})+2K\|\boldsymbol{\beta}\|_{\infty}\max_{1\leq
i\leq d}(\|w^{(i)}-e\|_{\infty})$
$\displaystyle=\psi(\xi)(\hat{w})+2K\|\boldsymbol{\beta}\|_{\infty}(\|w-e\|_{\infty}).$

∎

######  Lemma 6.6.

For any $\xi\in\prod_{i=1}^{d}\mathbbm{D}^{\beta_{i}}[0,T]$,

$\|\psi(\xi(w^{(1)}),\ldots,\xi(w^{(d)}))-\psi(\xi^{(1)},\ldots,\xi^{(d)})\|_{\infty}\leq
3K\|\boldsymbol{\beta}\|_{\infty}\|w-e\|_{\infty}.$

###### Proof.

Due to Lemma 6.4 and Lemma 6.5,

$\displaystyle\psi(\xi(w^{(1)}),\ldots,\xi(w^{(d)}))-\psi(\xi^{(1)},\ldots,\xi^{(d)})$
$\displaystyle\leq\psi(\xi)(\check{w})-\psi(\xi^{(1)},\ldots,\xi^{(d)})+2K\|\boldsymbol{\beta}\|_{\infty}(\|w-e\|_{\infty})$
$\displaystyle\leq\psi(\xi)+K\|\boldsymbol{\beta}\|_{\infty}(\|w-e\|_{\infty})-\psi(\xi)+2K\|\boldsymbol{\beta}\|_{\infty}(\|w-e\|_{\infty})$
$\displaystyle=3K\|\boldsymbol{\beta}\|_{\infty}(\|w-e\|_{\infty}).$

For the other inequality, notice that

$\psi(\xi^{(1)},\ldots,\xi^{(d)})-\psi(\xi(w^{(1)}),\ldots,\xi(w^{(d)}))\leq\psi(\xi)-\psi(\xi)(\hat{w})+2K\|\boldsymbol{\beta}\|_{\infty}\|w-e\|_{\infty}\leq
3K\|\boldsymbol{\beta}\|_{\infty}\|w-e\|_{\infty}.$

∎

####  6.2.2 Lipschitz continuity of the reflection map

In this section, we prove the Lipschitz continuity of the regulator map and
the buffer content component map in the product $J_{1}$ topology. We start
with the Lipschitz continuity of the regulator map $\psi$.

###### Proof of Proposition 2.1.

Given $\xi$ and $\zeta$, let $\delta$ be such that $d_{p}(\xi,\zeta)<\delta$.
Then, there exists $\lambda^{(i)}$ s.t.
$\|\xi^{(i)}\circ\lambda^{(i)}-\zeta^{(i)}\|_{\infty}\vee\|\lambda^{(i)}-e\|_{\infty}<\delta$
for each $i=1,\ldots,d$. Notice that

$\displaystyle d_{p}(\psi(\xi),\psi(\zeta))$
$\displaystyle\leq\inf_{w^{(1)},\ldots,w^{(d)}\in\Lambda}\left\\{\sum_{i=1}^{d}\|\psi^{(i)}(\xi)\circ
w^{(i)}-\psi^{(i)}(\zeta)\|_{\infty}\right\\}\leq\sum_{i=1}^{d}\|\psi^{(i)}(\xi)\circ\lambda^{(1)}-\psi^{(i)}(\zeta)\|_{\infty}$
$\displaystyle\leq\sum_{i=1}^{d}\|\psi^{(i)}(\xi)\circ\lambda^{(1)}-\psi^{(i)}(\xi^{1}\circ\lambda^{(1)},\ldots,\xi^{(d)}\circ\lambda^{(d)})\|_{\infty}$
$\displaystyle\qquad\qquad+\sum_{i=1}^{d}\|\psi^{(i)}(\xi^{1}\circ\lambda^{(1)},\ldots,\xi^{(d)}\circ\lambda^{(d)})-\psi^{(i)}(\zeta^{(1)},\ldots,\zeta^{(d)})\|_{\infty}$
$\displaystyle\leq\sum_{i=1}^{d}\|\psi^{(i)}(\xi)\circ\lambda^{(1)}-\psi^{(i)}(\xi^{1}\circ\lambda^{(1)},\ldots,\xi^{(d)}\circ\lambda^{(d)})\|_{\infty}+d\cdot
K\max_{1\leq i\leq d}\|\xi^{(i)}\circ\lambda^{(i)}-\zeta^{(i)}\|_{\infty}$
$\displaystyle\leq\sum_{i=1}^{d}\|\psi^{(i)}(\xi)\circ\lambda^{(1)}-\psi^{(i)}(\xi^{1}\circ\lambda^{(1)},\ldots,\xi^{(d)}\circ\lambda^{(d)})\|_{\infty}+d\cdot
K\cdot\delta/2$
$\displaystyle=\sum_{i=1}^{d}\|\psi^{(i)}(\xi)-\psi^{(i)}(\xi^{1}\circ\lambda^{(1)},\ldots,\xi^{(d)}\circ\lambda^{(d)})\circ{\lambda^{(1)}}^{-1}\|_{\infty}+d\cdot
K\cdot\delta/2$
$\displaystyle=\sum_{i=1}^{d}\|\psi^{(i)}(\xi)-\psi^{(i)}(\xi^{1}\circ\lambda^{(1)}\circ{\lambda^{(1)}}^{-1},\ldots,\xi^{(d)}\circ\lambda^{(d)}\circ{\lambda^{(1)}}^{-1})\|_{\infty}+d\cdot
K\cdot\delta/2.$ (6.3)

Since $\|\lambda^{(i)}\circ{\lambda^{(1)}}^{-1}-e\|<2\delta$ for each
$i=1,\ldots,d$, and by invoking Lemma 6.6, we have that

$\|\psi(\xi)-\psi(\xi^{1}\circ\lambda^{(1)}\circ{\lambda^{(1)}}^{-1},\ldots,\xi^{(d)}\circ\lambda^{(d)}\circ{\lambda^{(1)}}^{-1})\|_{\infty}\leq
d6K\|\boldsymbol{\beta}\|_{\infty}\delta.$ (6.4)

Combining (6.2.2) and (6.4), we have that $d_{p}(\psi(\xi),\psi(\zeta))\leq
Kd(6\|\boldsymbol{\beta}\|_{\infty}+1)\delta.$ Letting $\delta\downarrow
d_{p}(\xi,\zeta)$ we obtain Lipschitz continuity of $\psi$. ∎

###### Proof of Theorem 2.1.

The Lipschitz continuity of the regulator map has been proven in Proposition
2.1. We only need to verify the Lipschitz continuity of the buffer content
component map $\phi$. Let $\delta$ be such that $d_{p}(\xi,\zeta)<\delta$.
Then, $d_{J_{1}}(\xi^{(i)},\zeta^{(i)})<\delta$ for each $i=1\ldots,d$. Note
that
$\phi^{(i)}(\xi)=\xi^{(i)}+\psi^{(i)}(\xi)-\sum_{\\{j\in\\{1,\ldots,d\\}\setminus
i\\}}q_{ji}\psi^{(j)}\big{(}\xi\big{)}$. Hence,

$\displaystyle d_{J_{1}}(\phi^{(i)}(\xi),\phi^{(i)}(\zeta))$
$\displaystyle\leq
d_{J_{1}}(\xi^{(i)},\zeta^{(i)})+\sum_{i=1}^{d}d_{J_{1}}(\psi^{(i)}(\xi),\psi^{(i)}(\zeta))$
$\displaystyle<\delta+Kd(6\|\boldsymbol{\beta}\|_{\infty}+1)\delta=\delta(1+Kd(6\|\boldsymbol{\beta}\|_{\infty}+1)).$

Since
$d_{p}(\phi(\xi),\phi(\zeta))\leq\sum_{i=1}^{d}d_{J_{1}}(\phi^{(i)}(\xi),\phi^{(i)}(\zeta))\leq\delta
d(1+Kd(6\|\boldsymbol{\beta}\|_{\infty}+1))$, we have that $\phi$ is Lipschitz
continuous in $\prod_{i=1}^{d}\mathbbm{D}^{\beta_{i}}[0,T]$ by letting
$\delta\downarrow d_{p}(\xi,\zeta)$. ∎

##  Appendix A Continuity of some useful functions

In this appendix, we include the proofs of continuity of some functions in the
product $J_{1}$ topology. Recall the function
$\Upsilon_{\kappa}:\prod_{i=1}^{d}\mathbbm{D}[0,T]\to\prod_{i=1}^{d}\mathbbm{D}[0,T]$
where $\Upsilon_{\kappa}(\xi)(t)=\xi(t)-\kappa\cdot t$ for $t\in[0,T]$ and
$\pi:\prod_{i=1}^{d}\mathbbm{D}[0,T]\to\mathbb{R}^{d}$ where
$\pi(\xi)=\xi(T)$.

###### Proof of Lemma 2.2..

For i), suppose that $d_{p}(\xi,\zeta)<\epsilon$ w.r.t. the product $J_{1}$
topology. Then, there exists a homeomorphism $\lambda^{(i)}$,
$i\in\\{1,\ldots,d\\}$, so that
$\|\xi-\zeta\circ\lambda^{(i)}\|_{\infty}+\|\lambda^{(i)}-e(\cdot)\|_{\infty}<2\epsilon.$
Therefore,

$\displaystyle
d_{J_{1}}\left(\Upsilon_{\kappa}^{(i)}(\xi),\Upsilon_{\kappa}^{(i)}(\zeta)\right)$
$\displaystyle\leq\|\xi^{(i)}-\zeta^{(i)}\circ\lambda^{(i)}+\kappa^{(i)}(\lambda^{(i)}-e(\cdot))\|_{\infty}+\|\lambda^{(i)}-e(\cdot)\|_{\infty}$
$\displaystyle\leq\|\xi^{(i)}-\zeta^{(i)}\circ\lambda^{(i)}\|_{\infty}+(1+|\kappa^{(i)}|)\|\lambda^{(i)}-e(\cdot)\|_{\infty}$
$\displaystyle\leq 2(1+|\kappa^{(i)}|)\epsilon.$ (A.1)

Consequently,

$d_{p}(\Upsilon_{\kappa}(\zeta),\Upsilon_{\kappa}(\xi))\leq\sum_{i=1}^{d}d_{p}(\Upsilon^{(i)}_{\kappa}(\zeta),\Upsilon^{(i)}_{\kappa}(\xi))\leq
2\sum_{i=1}^{d}(1+|\kappa^{(i)}|)\epsilon=2(d+\sum_{i=1}^{d}|\kappa^{(i)}|)\epsilon.$

ii). Note that
$\Upsilon_{\kappa}^{-1}(\zeta)(\cdot)=\zeta(\cdot)+\kappa(\cdot)=\Upsilon_{-\kappa}(\zeta)$.
Hence, $\Upsilon_{\kappa}$ is injective and surjective. Furthermore, the
continuity of $\Upsilon_{\kappa}^{-1}$ is obtained by applying i) to
$\Upsilon_{-\kappa}$. ∎

Finally, we prove that the projection map is Lipschitz continuous in the
product $J_{1}$ topology.

###### Proof of Lemma 2.3..

Suppose that $d_{p}(\xi,\zeta)\leq\epsilon$. Then, we have that
$d_{J_{1}}(\xi^{(i)},\zeta^{(i)})\leq\epsilon$ for every $i=1,\ldots,d$. For
any homeomorphism $\lambda:[0,T]\to[0,T]$ that satisfies $\lambda(0)=0$ and
$\lambda(T)=T$, we have that

$|\pi^{(i)}(\xi)-\pi^{(i)}(\zeta)|=|\xi^{(i)}(T)-\zeta^{(i)}(T)|=|\xi^{(i)}(T)-\zeta^{(i)}(\lambda(T))|\leq\|\xi^{(i)}-\zeta^{(i)}\circ\lambda\|_{\infty}.$

Since this is true for any $\lambda$, $|\pi^{(i)}(\xi)-\pi^{(i)}(\zeta)|\leq
d_{J_{1}}(\xi^{(i)},\zeta^{(i)})\leq\epsilon.$ Since this holds for every
$i=1,\ldots,d,$ we have that $\|\pi(\xi)-\pi(\zeta)\|_{\infty}\leq\epsilon$. ∎

##  Appendix B Some useful tools on large deviations

In this appendix, we include results that facilitate the use of the extended
LDP. Given that the probability measures of $(X_{n})$ satisfy the extended LDP
in a metric space $\left(\mathcal{X},d\right)$, our results include the
derivation of the extended LDP in closed subspaces of $\mathcal{X}$, and a
variation of the contraction principle for Lipschitz continuous maps. Let
$\mathcal{D}_{I}\triangleq\\{x\in X:I(x)<\infty\\}$ denote the effective
domain of $I$.

######  Lemma B.1.

Let $E$ be a closed subset of $\mathcal{X}$ and let $X_{n}$ be such that
$\mathbf{P}(X_{n}\in E)=1$ for all $n\geq 1$. Suppose that $E$ is equipped
with the topology induced by $\mathcal{X}$. Then, if the probability measures
of $(X_{n})$ satisfy the extended LDP in $(\mathcal{X},d)$ with speed $a_{n}$,
and with rate function $I$ so that $\mathcal{D}_{I}\subset E$, then the same
extended LDP holds in $E$.

###### Proof.

In the topology induced on $E$ by $\mathcal{X}$, the open sets are sets of the
form $G\cap E$ with $G\subseteq\mathcal{X}$ open. Similarly, the closed sets
in this topology are the sets of the form $F\cap E$ with
$F\subseteq\mathcal{X}$ closed. Furthermore,
$\mathbf{P}(X_{n}\in\Gamma)=\mathbf{P}(X_{n}\in\Gamma\cap E)$ for any
$\Gamma\in\mathcal{B}$, where $\mathcal{B}$ is the Borel sigma-algebra.
Suppose that an extended LDP holds in $\mathcal{X}$. Now, for the upper bound,
let $F$ be a closed subset of $E$. Then, $F$ is a closed subset of
$\mathcal{X}$. Hence,
$\limsup_{n\to\infty}\frac{1}{a_{n}}\log\mathbf{P}\left(X_{n}\in
F\right)\leq-\inf_{x\in F^{\epsilon}}I(x)=-\inf_{x\in F^{\epsilon}\cap
E}I(x).$ Next, let $G$ be an open subset of $E$. That is, $G=G^{\prime}\cap
E$, where $G^{\prime}$ is an open subset of $\mathcal{X}$. Then,

$\displaystyle\liminf_{n\to\infty}\frac{1}{a_{n}}\log\mathbf{P}\left(X_{n}\in
G\right)=\liminf_{n\to\infty}\frac{1}{a_{n}}\log\mathbf{P}\left(X_{n}\in
G^{\prime}\cap E\right)$
$\displaystyle=\liminf_{n\to\infty}\frac{1}{a_{n}}\log\mathbf{P}\left(X_{n}\in
G^{\prime}\right)\geq-\inf_{x\in G^{\prime}\cap E}I(x)=-\inf_{x\in G}I(x).$

The level sets $\Psi_{I}(\alpha)\subseteq\mathcal{X}$ are closed, so $I$
restricted to $E$ remains lower semicontinuous. ∎

We continue with a useful lemma on pre-images of Lipschitz continuous maps on
metric spaces. For a closed subset of the metric space
$\left(\mathcal{X},d\right)$, recall that
$A^{\epsilon}\triangleq\\{\xi\in\mathcal{X}:d(\xi,A)\leq\epsilon\\}$, where
$d(\xi,A)=\inf_{\zeta\in A}d(\xi,\zeta)$.

######  Lemma B.2.

Let $(\mathbb{S},\sigma)$ and $(\mathbbm{X},d)$ be metric spaces. Suppose that
$\mathrm{\Phi}:\left(\mathbbm{X},d\right)\to\left(\mathbb{S},\sigma\right)$ is
a Lipschitz continuous mapping with Lipschitz constant
$\|\mathrm{\Phi}\|_{\mathrm{Lip}}$. Then, for any set $F\subset\mathbbm{X}$,
it holds that

$\left(\mathrm{\Phi}^{-1}(F)\right)^{\epsilon}\subseteq\mathrm{\Phi}^{-1}\left(F^{\|\mathrm{\Phi}\|_{\mathrm{Lip}}\epsilon}\right).$

###### Proof.

Let $\zeta\in\left(\mathrm{\Phi}^{-1}(F)\right)^{\epsilon}$. For each $n\geq
1$, since $\mathrm{\Phi}^{-1}(F)$ is a closed set, there exists a
$\xi_{n}\in\mathrm{\Phi}^{-1}(F)$ so that $d(\xi_{n},\zeta)\leq\epsilon+1/n$.
Since $\xi_{n}\in\mathrm{\Phi}^{-1}(F)$ we have that
$\mathrm{\Phi}(\xi_{n})\in F$. Furthermore,
$\sigma(\mathrm{\Phi}(\xi_{n}),\mathrm{\Phi}(\zeta))\leq\|\mathrm{\Phi}\|_{\mathrm{Lip}}(\epsilon+1/n)$.
Hence,
$d\big{(}\mathrm{\Phi}(\zeta),F\big{)}\leq\|\mathrm{\Phi}\|_{\mathrm{Lip}}(\epsilon+1/n)$.
Letting $n\to\infty$, we get
$d\big{(}\mathrm{\Phi}(\zeta),F\big{)}\leq\|\mathrm{\Phi}\|_{\mathrm{Lip}}\epsilon$,
that is, $\zeta\in\mathrm{\Phi}^{-1}(F^{\|\mathrm{\Phi}\|\epsilon})$. Since
this holds for any $\zeta\in\big{(}\mathrm{\Phi}^{-1}(F)\big{)}^{\epsilon}$,
the statement holds true. ∎

The following lemma is a version of the contraction principle adapted to the
setting of extended LDP’s.

######  Lemma B.3.

Let $(\mathbb{S},\sigma)$ and $(\mathbbm{X},d)$ be metric spaces. Suppose that
the sequence of probability measures of $(\mathbf{X}_{n})$ satisfies the
extended LDP in $(\mathbbm{X},d)$ with speed $a_{n}$ and rate function $I$.
Moreover, let
$\mathrm{\Phi}:\left(\mathbbm{X},d\right)\to\left(\mathbb{S},\sigma\right)$ be
a Lipschitz continuous mapping and set
$I^{\prime}(y)\triangleq\inf_{\mathrm{\Phi}(x)=y}I(x).$ Then,

  * i)

$\mathbf{S}_{n}=\mathrm{\Phi}(\mathbf{X}_{n})$ satisfies the following bounds:
for any open set $G\subseteq\mathbb{S}$,

$\liminf_{n\to\infty}\frac{1}{a_{n}}\log\mathbf{P}\left(\mathbf{S}_{n}\in
G\right)\geq-\inf_{x\in G}I^{\prime}(x),$

and for any closed set $F\subseteq\mathbb{S}$,

$\limsup_{n\to\infty}\frac{1}{a_{n}}\log\mathbf{P}\left(\mathbf{S}_{n}\in
F\right)\leq-\lim_{\epsilon\to 0}\inf_{x\in F^{\epsilon}}I^{\prime}(x).$

  * ii)

Suppose in addition that $\mathrm{\Phi}$ is a homeomorphism. Then,
$\mathbf{S}_{n}$ satisfies the extended LDP in $(\mathbb{S},\sigma)$ with
speed $a_{n}$ and rate function $I^{\prime}$.

  * iii)

If $I^{\prime}$ is a good rate function—i.e.,
$\Psi_{I^{\prime}}(M)\triangleq\\{y\in\mathbb{S}:I^{\prime}(y)\leq M\\}$ is
compact for each $M\in[0,\infty)$, then ${\mathbf{S}}_{n}$ satisfies the large
deviation principle in $(\mathbb{S},d)$ with speed $a_{n}$ and good rate
function $I^{\prime}$.

###### Proof.

i). For the upper bound, let $F$ be a closed subset of
$\left(\mathbb{S},\sigma\right)$. Thanks to Lemma B.2, for any $\epsilon>0$,
we have that
$\left(\mathrm{\Phi}^{-1}(F)\right)^{\epsilon}\subseteq\mathrm{\Phi}^{-1}\left(F^{\|\mathrm{\Phi}\|_{\mathrm{Lip}}\epsilon}\right)$.
Hence,

$-\inf_{x\in\left(\mathrm{\Phi}^{-1}(F)\right)^{\epsilon}}I(x)\leq-\inf_{x\in\mathrm{\Phi}^{-1}\left(F^{\epsilon\|\mathrm{\Phi}\|_{\mathrm{Lip}}}\right)}I(x).$
(B.1)

Furthermore, by the extended LDP of $\mathbf{X}_{n}$, for $\delta>0$ there
exists an $n(\delta)$ such that for any $n\geq n(\delta),$

$\mathbf{P}(\mathbf{X}_{n}\in\mathrm{\Phi}^{-1}(F))\leq\exp\left[a_{n}\left(-\inf_{x\in\left(\mathrm{\Phi}^{-1}(F)\right)^{\epsilon}}I(x)+\delta\right)\right]\
\text{for any }\ \epsilon>0.$ (B.2)

Consequently, (B.1), and (B.2) lead to

$\mathbf{P}(\mathbf{X}_{n}\in\mathrm{\Phi}^{-1}(F))\leq\exp\left[a_{n}\left(-\inf_{x\in\mathrm{\Phi}^{-1}\left(F^{\epsilon\|\mathrm{\Phi}\|_{\mathrm{Lip}}}\right)}I(x)+\delta\right)\right],$
(B.3)

$\text{for any}\ n\geq n(\delta)\ \text{and}\ \epsilon>0$. Next, for $n\geq
n(\delta)$,

$\mathbf{P}\left({\mathbf{S}}_{n}\in
F\right)=\mathbf{P}\left(\mathrm{\Phi}\left({\mathbf{X}}_{n}\right)\in
F\right)=\mathbf{P}\left({\mathbf{X}}_{n}\in\mathrm{\Phi}^{-1}\big{(}F\big{)}\right)\leq\exp\left[a_{n}\left(-\inf_{x\in\mathrm{\Phi}^{-1}\left(F^{\epsilon\|\mathrm{\Phi}\|_{\mathrm{Lip}}}\right)}I(x)+\delta\right)\right].$

Therefore,

$\limsup_{n\rightarrow\infty}\frac{1}{a_{n}}\log\mathbf{P}\left({\mathbf{S}}_{n}\in
F\right)\leq-\inf_{x\in\mathrm{\Phi}^{-1}\left(F^{\epsilon\|\mathrm{\Phi}\|_{\mathrm{Lip}}}\right)}I(x)+\delta=-\inf_{y\in
F^{\epsilon\|\mathrm{\Phi}\|_{\mathrm{Lip}}}}I^{\prime}(y)+\delta.$ (B.4)

Letting $\delta\to 0$ and $\epsilon\to 0$ in (B.4), we arrive at the desired
large deviation upper bound.

Turning to the lower bound, consider an open set $G$. Since
$\mathrm{\Phi}^{-1}(G)$ is open,

$\liminf_{n\rightarrow\infty}\frac{1}{a_{n}}\log\mathbf{P}\left(\mathbf{S}_{n})\in
G\right)=\liminf_{n\rightarrow\infty}\frac{1}{a_{n}}\log\mathbf{P}\left({\mathbf{X}}_{n}\in\mathrm{\Phi}^{-1}(G)\right)\geq-\inf_{y\in\mathrm{\Phi}^{-1}\left(G\right)}I(\xi)=-\inf_{x\in
G}I^{\prime}(x).$

ii). The upper and lower bounds for the extended large deviation principle
have been proved in $i)$. Since $I$ is a rate function, its level sets
$\Psi_{I}(M)\triangleq\\{x\in\mathbb{X}:I(x)\leq M\\}$ are closed for every
$M>0$. Now, we verify that $I^{\prime}$ is lower-semicontinuous. The level
sets of $I^{\prime}$ are
$\Psi_{I^{\prime}}(M)\triangleq\\{y\in\mathbb{S}:I^{\prime}(y)\leq M\\}$, for
every $M>0$ . Note that

$\\{y\in\mathbb{S}:I^{\prime}(y)\leq M\\}=\\{\mathrm{\Phi}(x):I(x)\leq
M\\}=\mathrm{\Phi}(\Psi_{I}(M)).$

Since $\mathrm{\Phi}$ is a homeomorphism the r.h.s. is closed. Hence,
$\mathbf{S}_{n}$ satisfies the extended LDP.
iii). Due to our assumption, $I^{\prime}$ is a good rate function. Applying
$i)$, we obtain $\lim_{\epsilon\to 0}\inf_{y\in
F^{\|\mathrm{\Phi}\|_{\mathrm{Lip}}\epsilon}}I^{\prime}(y)=\inf_{y\in
F}I^{\prime}(y).$ Consequently,

$\displaystyle\limsup_{n\rightarrow\infty}\frac{\log\mathbf{P}\left({\mathbf{S}}_{n}\in
F\right)}{a_{n}}\leq-\lim_{\epsilon\to 0}\inf_{y\in
F^{\epsilon\|\mathrm{\Phi}\|_{\mathrm{Lip}}}}I^{\prime}(y)=-\inf_{y\in
F}I^{\prime}(y).$

∎

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