# Strong convergence to two-dimensional alternating Brownian motion processes

Guy Latouche  Department of Computer Science, Université Libre de Bruxelles
Giang T. Nguyen  School of Mathematical Sciences, The University of Adelaide
Oscar Peralta  School of Mathematical Sciences, The University of Adelaide

###### Abstract

Flip-flop processes refer to a family of stochastic fluid processes which
converge to either a standard Brownian motion (SBM) or to a Markov modulated
Brownian motion (MMBM). In recent years, it has been shown that complex
distributional aspects of the univariate SBM and MMBM can be studied through
the limiting behaviour of flip-flop processes. Here, we construct two classes
of bivariate flip-flop processes whose marginals converge strongly to SBMs and
are dependent on each other, which we refer to as _alternating two-dimensional
Brownian motion processes_. While the limiting bivariate processes are not
Gaussian, they possess desirable qualities, such as being tractable and having
a time-varying correlation coefficient function.

Strong convergence, two-dimensional Brownian motion, stochastic fluid
processes, flip-flops, time-varying correlation coefficient

_2010 Mathematics Subject Classification:_ Primary: 60F15, 60G50; Secondary:
60G15

##  1 Introduction

Let $\mathcal{B}=\\{B(t)\\}_{t\geq 0}$ denote a standard Brownian motion. The
construction of discrete-time random walk approximations to $\mathcal{B}$ is a
well-known problem initiated by Donsker in [10]. His work dealt with weak
approximations, that is, with the convergence of the _laws_ of appropriately
scaled random walks to the law of $\mathcal{B}$. A different technique using
strong approximations was proposed by Strassen [24], which involved the
construction of a probability space where the paths of the scaled random walks
were shown to converge uniformly on compact intervals to the path of
$\mathcal{B}$. Not only did Strassen’s technique provide an alternative
approach to the construction of approximations to $\mathcal{B}$, but it also
yielded stronger results: a strongly convergent family of stochastic processes
to $\mathcal{B}$ is automatically weakly convergent, while the opposite is not
always true. Furthermore, strong convergence allows one to claim convergence
of pathwise functionals of the processes.

In this paper we focus on strong approximations to $\mathcal{B}$ and related
models, not using discrete-time random walks, but a family of continuous-time
stochastic processes which we introduce next. Define a family of processes
$\\{\mathcal{F}_{\lambda}\\}_{\lambda>0}=\\{\\{F_{\lambda}(t)\\}_{t\geq
0}\\}_{\lambda>0}$, where

$\displaystyle
F_{\lambda}(t)=\sqrt{\lambda}\int_{0}^{t}J_{\lambda}(s)\mathrm{d}s,\quad t\geq
0,$

and the _phase_ process $\mathcal{J}_{\lambda}=\\{J_{\lambda}(t)\\}_{t\geq 0}$
denotes a Markov jump process with state space $\mathcal{E}=\\{1,-1\\}$ and
intensity matrix given by

$\displaystyle\Lambda=\left[\begin{array}[]{rr}-\lambda&\lambda\\\
\lambda&-\lambda\end{array}\right].$ (1.3)

The process $\mathcal{F}_{\lambda}$ describes the path of a particle in
$\mathds{R}$ with finite constant velocity which randomly switches direction
at a constant rate. Such a process has been studied in the literature as the
telegraph process [11, 14], the uniform transport process [25, 13], the flip-
flop process [23, 17], though in most instances $\mathcal{F}_{\lambda}$ is a
particular example of a more general model. Here we will refer to
$\mathcal{F}_{\lambda}$ as the _standard flip-flop process_.

The standard flip-flop process $\mathcal{F}_{\lambda}$ belongs to a larger
class of piecewise-linear models modulated by a Markov jump process, called
_stochastic fluid processes_ (SFP), for which several first passage results
and formulae are available [15]. Even if the behaviour of
$\mathcal{F}_{\lambda}$ is somewhat similar to that of a random walk with
double-sided exponential increments, notice that changes in direction of
$\mathcal{F}^{\lambda}$ happen at random time epochs. In comparison, the
discrete-time random walks approximations to $\mathcal{B}$ devised in [10, 24]
have inflection points on a deterministic grid. Though this characteristic of
$\mathcal{F}_{\lambda}$ may seem inconvenient at first, once a limiting result
is established, it actually allows us to translate well-understood results
from SFPs into complex second-order models such as the Markov modulated
Brownian motion (MMBM), as recently evidenced in [17, 16, 2, 18, 1, 20].
Besides being important objects of study within the matrix-analytic-methods
community, both SFPs and MMBMs have been succesfully used to model random
systems in queueing theory, risk theory, finance and epidemiology.

The _weak_ convergence of $\mathcal{F}_{\lambda}$ to $\mathcal{B}$ as
$\lambda\rightarrow\infty$ was first established in [25]. Through different
constructions, it was shown in [13, 21, 22] that $\mathcal{F}_{\lambda}$ also
converges _strongly_ to $\mathcal{B}$; that is, they showed the existence of a
probability space $(\Omega,\mathscr{F},\mathds{P})$ such that for all $T\geq
0$,

$\displaystyle\lim_{\lambda\rightarrow\infty}\sup_{0\leq s\leq
T}|F_{\lambda}(s)-B(s)|=0\quad\mbox{almost surely}.$ (1.4)

In order to extend these strong approximations into two-dimensional processes,
consider two families of standard flip-flops,
$\\{\mathcal{F}^{1}_{\lambda}\\}$ and $\\{\mathcal{F}^{2}_{\lambda}\\}$, with
corresponding families of phase processes $\\{\mathcal{J}_{\lambda}^{1}\\}$
and $\\{\mathcal{J}_{\lambda}^{2}\\}$, living on two probability spaces
$(\Omega^{1},\mathscr{F}^{1},\mathds{P}^{1})$ and
$(\Omega^{2},\mathscr{F}^{2},\mathds{P}^{2})$, respectively. One can see that,
on
$(\Omega^{1}\times\Omega^{2},\mathscr{F}^{1}\otimes\mathscr{F}^{2},\mathds{P}^{1}\times\mathds{P}^{2})$,
the bivariate process $(\mathcal{F}^{1}_{\lambda},\mathcal{F}^{2}_{\lambda})$
converges strongly to a two-dimensional Brownian motion
$(\mathcal{B}^{1},\mathcal{B}^{2})$ as $\lambda\rightarrow\infty$.

The process $\left(\mathcal{F}^{1}_{\lambda},\mathcal{F}^{2}_{\lambda}\right)$
can be redefined such that $\mathcal{F}^{1}_{\lambda}$ and
$\mathcal{F}^{2}_{\lambda}$ are jointly modulated by a common phase process
$\bm{\mathcal{J}}_{\lambda}=\\{\bm{J}_{\lambda}(t)\\}=\left\\{\left(J_{\lambda}^{1}(t),J_{\lambda}^{2}(t)\right)\right\\}$
on state space $\mathcal{E}^{2}$, as follows: for $i=1,2$,

$\displaystyle
F^{i}_{\lambda}(t)=\sqrt{\lambda}\int_{0}^{t}\pi_{i}\left(\bm{J}_{\lambda}(s)\right)\mathrm{d}s,\quad
t\geq 0,$

where $\pi_{i}:\mathcal{E}^{2}\mapsto\mathcal{E}$ denotes the $i$th coordinate
projection, and $\bm{\mathcal{J}}_{\lambda}$ is a Markov jump process with the
intensity matrix

$\displaystyle\Lambda\oplus\Lambda=\left[\begin{array}[]{rrrr}-2\lambda&\lambda&\lambda&0\\\
\lambda&-2\lambda&0&\lambda\\\ \lambda&0&-2\lambda&\lambda\\\
0&\lambda&\lambda&-2\lambda\end{array}\right],$ (1.9)

where $\oplus$ denotes the Kronecker sum and
$\mathcal{E}^{2}=\\{(1,1),(1,-1),(-1,1),(-1,-1)\\}$. We refer to
$\left(\mathcal{F}^{1}_{\lambda},\mathcal{F}^{2}_{\lambda}\right)$ as a
_standard bivariate flip-flop_ process, and $\bm{\mathcal{J}}_{\lambda}$ its
underlying phase process. More broadly, we define the following.

######  Definition 1.1.

A _bivariate flip-flop_ process
$\left(\mathcal{G}^{1}_{\lambda},\mathcal{G}^{2}_{\lambda}\right)=\left\\{\left({G}^{1}_{\lambda}(t),{G}^{2}_{\lambda}(t)\right)\right\\}_{t\geq
0}$ is defined by

$\displaystyle
G^{i}_{\lambda}(t)=\sqrt{\lambda}\int_{0}^{t}\pi_{i}(J_{\lambda}(s))\mathrm{d}s,\quad
t\geq 0,$ (1.10)

where $\mathcal{J}_{\lambda}$ is a Markov jump process on
$\mathcal{E}^{2}\times\mathcal{S}$ (where $\mathcal{S}$ is some finite
collection of states), and
$\pi_{i}:\mathcal{E}^{2}\times\mathcal{S}\rightarrow\mathds{R}$ is such that
$\pi_{i}(\\{1\\}\times\mathcal{S})\subset(0,\infty)$ and
$\pi_{i}(\\{-1\\}\times\mathcal{S})\subset(-\infty,0)$.

Thus, a bivariate flip-flop process has an underlying phase process
$\bm{\mathcal{J}}_{\lambda}$ with a considerably richer structure than that of
(1.9), which may be used to produce dependent coordinate processes instead of
independent ones. One natural goal is to characterise all possible strong
limits for bivariate flip-flop processes and study their dependence
structures.

As a step toward that goal, we construct in this paper two families of
bivariate flip-flop processes and show that each converges strongly to a
process, of which the marginals are standard Brownian motion and the
correlation between the two components is a time-varying function. In short,
the marginals of the limiting process alternate between _synchronising_
intervals during which they evolve with identical increments to each other,
and _desynchronising_ intervals during which they evolve as mirror images of
each other. In Section 3 the intervals are exponentially distributed and we
call the limiting process a _two-dimensional exponentially alternating
Brownian motion_. In Section 4 the synchronisation and desynchronisation
epochs are determined by a continuous-time Markovian arrival process (MAP); we
refer to this process as a _two-dimensional MAP alternating Brownian motion_.
We determine the time-varying correlation functions of the limiting processes
in Sections 3 and 4, respectively.

From a different perspective, the bivariate flip-flop processes can be seen as
a continuous-time analogue of the discrete-time _bootstrapping random walk_
concept proposed in [9, 8]. There, the authors start by constructing a
discrete-time random walk from Bernoulli random variables, and then reuse
(bootstrap) these random variables to create additional walks; together, the
random walks converge weakly to two- or higher-dimensional Brownian motions.
In our case, we will start by constructing a continuous-time random walk
(standard flip-flop) from exponential random variables, and then reuse them to
create an additional continuous-time random walk; together, the random walks
(bivariate flip-flop) converge strongly to a two-dimensional process. In one
construction, the strong limit is a two-dimensional Brownian motion; in two
others, the strong limits are two-dimensional alternating Brownian motion
processes.

Another related model is the zig-zag process [6], a multidimensional piecewise
deterministic Markov process which behaves in a piecewise linear fashion over
time. Changes in direction of the path of a zig-zag process occur according to
some space-dependent intensity function, the latter chosen in such a way that
the stationary behaviour of the process follows some prespecified
distribution. Although flip-flop processes are in fact a space-homogeneous
version of the zig-zag processes, their use differ: zig-zag processes are
commonly used to implement efficient sampling methods [4, 5], while the goal
of flip-flop processes is to explain properties of complex second-order
processes.

##  2 Preliminaries

Our construction of bivariate flip-flops is inspired by [21], and we sketch in
this section the construction proposed there for one-dimensional standard
flip-flops. The creation in [21] of a family of standard flip-flops
$(\mathcal{F}_{\lambda},\mathcal{J}_{\lambda})$, with
$\lambda\rightarrow\infty$, relies on inspecting the standard Brownian motion
$\mathcal{B}$ at the arrival epochs of a Poisson process and sequentially
using the Wiener-Hopf factorisation between these observation times, as
explained next.

For $\lambda>0$, let $\mathcal{N}_{\lambda}$ be a Poisson process of parameter
$\lambda/2$, independent of $\mathcal{B}$, and denote by
$\\{\theta^{\lambda}_{k}\\}_{k\geq 0}$ its arrival epochs. For notational
convenience, we omit the superscript $\lambda$ when no ambiguity might arise.
For $k\geq 0$, let

$\displaystyle C_{k}=B(\theta_{k}),\quad
M_{k}=\min_{s\in(\theta_{k},\theta_{k+1})}B(s).$

Define the jump process $\mathcal{J}_{\lambda}$ with $J_{\lambda}(0)=-1$ and
jump epochs $\\{\xi_{k}^{\lambda}\\}_{k\geq 0}$ (or $\\{\xi_{k}\\}_{k\geq 0}$)
with $\xi_{0}=0$ such that

  * •

$\xi_{2k+1}-\xi_{2k}=\lambda^{-1/2}(C_{k}-M_{k})$ and $J(s)=-1$ for all
$s\in[\xi_{2k},\xi_{2k+1})$,

  * •

$\xi_{2k+2}-\xi_{2k+1}=\lambda^{-1/2}(C_{k+1}-M_{k})$ and $J(s)=1$ for all
$s\in[\xi_{2k+1},\xi_{2k+2})$.

By Wiener-Hopf factorisation for Brownian motion [3, p.165], the random
variables $(C_{k}-M_{k})$ and $(C_{k+1}-M_{k})$ are independent and
$\mbox{exp}(\sqrt{\lambda})$–distributed. Scaling these by $\lambda^{-1/2}$
yields alternating occupation times that are $\mbox{exp}(\lambda)$, which
implies that $\mathcal{J}_{\lambda}$ is a Markov jump process and its
intensity matrix $\Lambda$ is given by (1.3).

Intuitively, to see that $\mathcal{F}_{\lambda}$ approximates $\mathcal{B}$,
define the seequence $\\{\chi^{\lambda}_{k}\\}_{k\geq 0}$ (or
$\\{\chi_{k}\\}_{k\geq 0}$) by taking $\chi_{0}=0$ and $\chi_{k}=\xi_{2k}$.
This sequence $\\{\chi_{k}\\}_{k\geq 0}$ is one of two most important
sequences underlying the strong approximation, with $\\{\theta_{k}\\}_{k\geq
0}$ being the other. There are three key properties of the construction.
First,

$\displaystyle
F_{\lambda}(\chi_{k})=\sqrt{\lambda}\sum_{i=0}^{k-1}\left[\lambda^{-1/2}\left(M_{i}-C_{i}\right)+\lambda^{-1/2}\left(C_{i+1}-M_{i}\right)\right]=C_{k}-C_{0}=B(\theta_{k}).$

Second, the minima of $\mathcal{F}_{\lambda}$ within each interval
$(\chi_{k},\chi_{k+1})_{k\geq 0}$ match the values of $\\{M_{k}\\}_{k\geq 1}$.
Figure 1 illustrates a sample path of an SBM and the associated sample path of
$\mathcal{F}_{\lambda}$ constructed in the aforementioned procedure.

Figure 1: (Left) A sample path of an SBM $\mathcal{B}$. Arrivals corresponding
to $\mathcal{N}_{\lambda}$ occur at $\\{\theta_{k}\\}_{k\geq 0}$, and the
minima of $\mathcal{B}$, $\\{M_{k}\\}_{k\geq 1}$, attained between these
arrivals are highlighted with red crosses. (Right) An associated sample path
of a standard flip-flop $\mathcal{F}_{\lambda}$. The values of
$F_{\lambda}(\chi_{k})$ match those of $B(\theta_{k})$ for $k=1,2,3$,
respectively; the values of $F_{\lambda}(\xi_{2k+1})$ are the same as the
minima $\\{M_{k}\\}$, for $k=1,2,3,4$, respectively.

Third, since $\theta_{k}\sim\mbox{Erlang}(k,\lambda/2)$ and
$\chi_{k}\sim\mbox{Erlang}(2k,\lambda)$, we have
$\mathds{E}(\theta_{k})=\mathds{E}(\chi_{k})$, implying that on average the
epochs $\\{\theta_{k}\\}$ coincide with $\\{\chi_{k}\\}$. More precisely,
Nguyen and Peralta [21] shows that

$\displaystyle\lim_{\lambda\rightarrow\infty}\max_{k:\theta_{k}^{\lambda},\chi_{k}^{\lambda}<T}|\theta_{k}^{\lambda}-\chi_{k}^{\lambda}|$
$\displaystyle=0\quad\mbox{almost surely.}$

##  3 Reflection at exponential times

Here, we construct on a probability space $(\Omega,\mathscr{F},\mathds{P})$ a
family of bivariate flip-flops
$\\{(\mathcal{G}^{1}_{\lambda},\mathcal{G}^{2}_{\lambda})\\}_{\lambda\geq 0}$,
such that $\\{(\mathcal{G}^{1}_{\lambda},\mathcal{G}^{2}_{\lambda})\\}$
converges strongly to a so-called _two-dimensional exponentially alternating
Brownian motion_ as $\lambda\rightarrow\infty$.

######  Definition 3.1.

Let $\mathcal{X}=\\{X(t)\\}_{t\geq 0}$ be a Markov jump process on state space
$\\{0,1\\}$, with initial probability $(1,0)$ and intensity matrix

$\displaystyle\Pi=\left[\begin{array}[]{rr}-\alpha&\alpha\\\
\beta&-\beta\end{array}\right],$ (3.3)

and let $\mathcal{B}$ be a standard Brownian motion. Define
$\mathcal{B}^{*}=\\{B^{*}(t)\\}_{t\geq 0}$ such that

$\displaystyle B^{*}(t)=\int_{0}^{t}(-1)^{X(s)}\mathrm{d}B_{s},\quad t\geq 0.$
(3.4)

The process $(\mathcal{B},\mathcal{B}^{*})$ is called a two-dimensional
exponentially alternating Brownian motion.

In the intervals during which $X(t)=0$, the processes $\mathcal{B}$ and
$\mathcal{B}^{*}$ evolve with precisely the same increment; we refer to these
periods as _synchronising intervals_. In intervals during which $X(t)=1$, the
increment of one process is the negative of the other; these are
_desynchronising intervals_.

It is important to notice that $(\mathcal{B},\mathcal{B}^{*})$ is not a
Gaussian process: for instance, the distribution of $B(t)-B^{*}(t)$ has a
point mass at $0$ of size $e^{-\alpha t}$, corresponding to the event
$\\{X(s)=0\mbox{ for all }0\leq s\leq t\\}$. Despite not being Gaussian,
$(\mathcal{B},\mathcal{B}^{*})$ is still a process with tractable properties,
as we show later in Section 3.2.

###  3.1 Construction

Step 1. We start by constructing the Markov jump process $\mathcal{X}$ through
a process of uniformization. Fix $\alpha,\beta>0$ and let
$\gamma=\alpha+\beta$. Enlarge the probability space
$(\Omega,\mathscr{F},\mathds{P})$ so that it supports an independent discrete-
time Markov chain $\mathcal{Y}=\\{Y(m)\\}_{m\in\mathds{N}}$ with state space
$\\{0,1\\}$, initial probability $(1,0)$ and transition probability matrix

$\displaystyle P=I+\gamma^{-1}\Pi,$ (3.5)

where $I$ is the $2\times 2$ identity matrix. Notice that $P$ happens to have
identical rows, but this characteristic is not necessarily important.

Let $\\{\lambda_{n}\\}_{n\in\mathds{N}}$ be an increasing sequence such that
$\lambda_{0}=2\gamma$ and $\lim_{n\rightarrow\infty}\lambda_{n}=\infty$. Let
$\\{\mathcal{N}_{\lambda_{n}}\\}_{n\in\mathds{N}}$ be a nested family of
Poisson processes with intensities $\lambda_{n}/2$, where nested means the
events of $\mathcal{N}_{\lambda_{n+1}}$ result from the superposition of
$\mathcal{N}_{\lambda_{n}}$ and an additional, independent, Poisson process.
These processes are used in Step 2 but the slowest Poisson process,
$\mathcal{N}_{\lambda_{0}}$, is used to build the Markov jump process
$\mathcal{X}$ through $X(t)=Y(N_{\lambda_{0}}(t))$ for $t\geq 0.$ One easily
verifies that $\mathcal{X}$ is a Markov jump process on $\\{0,1\\}$ with
initial probability $(1,0)$ and intensity matrix given by $\eqref{eqn:eJH}$.

With a slight abuse of notation, let $\\{\theta_{k}^{n}\\}_{k\in\mathds{N}}$
be the arrival times of $\mathcal{N}_{\lambda_{n}}$, and denote by
$\\{\ell(k)\\}_{k\in\mathds{N}}$ the sequence of epochs at which the Markov
chain $\mathcal{Y}$ jumps to different states, that is, $\ell(0)=0$ and

$\displaystyle\ell(k)=\inf\\{m>\ell(k-1):Y(m)\neq Y(m-1)\\}.$ (3.6)

Then, by construction, the $k$th jump of the Markov jump process $\mathcal{X}$
occurs at time $\theta_{\ell(k)}^{0}$, and we refer to $\\{\ell(k)\\}$ as the
_switching epochs_ of $\mathcal{Y}$. This is illustrated on Figure 2: blue
dots mark the events of $\mathcal{N}_{\lambda_{0}}$, red diamonds those of
$\mathcal{N}_{\lambda_{1}}$, and black squares mark the jump epochs
$\theta_{\ell(k)}^{0}$ of $\mathcal{X}$.

00$\theta^{0}_{1}$$\theta^{0}_{2}$$\theta^{0}_{3}$$\theta^{0}_{4}$$\theta^{0}_{5}$$\theta^{0}_{6}$$\theta^{1}_{1}$$\theta^{1}_{2}$$\theta^{1}_{3}$$\theta^{1}_{4}$$\theta^{1}_{5}$$\theta^{1}_{6}$$\theta^{1}_{7}$$\theta^{1}_{8}$$\theta^{1}_{9}$$\theta^{1}_{10}$$\theta^{1}_{11}$

Figure 2: Blue dots correspond to arrivals $\\{\theta^{0}_{k}\\}_{k\geq 0}$ of
$\mathcal{N}_{\lambda_{0}}$, red diamonds the arrivals
$\\{\theta^{1}_{k}\\}_{k\geq 0}$ of $\mathcal{N}_{\lambda_{1}}$, black squares
the jump epochs of $\mathcal{X}$. Here, $\ell(1)=2,\ell(2)=3,$ and
$\ell(3)=5$.

Step 2. For each $n\in\mathds{N}$, we construct the phase process
$\bm{\mathcal{J}}_{\lambda_{n}}=(\mathcal{J}_{\lambda_{n}}^{1},\mathcal{J}_{\lambda_{n}}^{2})$
to modulate the bivariate flip-flop
$(\mathcal{G}_{\lambda_{n}}^{1},\mathcal{G}_{\lambda_{n}}^{2})$. Define a
discrete-time process $\mathcal{Y}_{n}=\\{Y_{n}(k)\\}_{k\in\mathds{N}}$ such
that

$\displaystyle Y_{n}(k)=X(\theta^{n}_{k}),\qquad k\in\mathds{N}.$ (3.7)

By the nested construction of the Poisson processes
$\\{\mathcal{N}_{\lambda_{n}}\\}$, for each $n\geq 1$ and $k\geq 0$ there
exists a _unique_ $\nu_{n}(k)\in\mathds{N}$ such that
$\theta_{\nu_{n}(k)}^{n}=\theta_{\ell(k)}^{0}$. In other words, the
$\nu_{n}(k)$-th arrival epoch of $\mathcal{N}_{\lambda_{n}}$ is the $k$th jump
epoch of $\mathcal{X}$; for example, in Figure 2,
$\nu_{1}(1)=2,\nu_{1}(2)=5,\nu_{1}(3)=8$. Importantly, the process
$\mathcal{Y}_{n}$, which lives on $\\{0,1\\}$, changes states _only_ at
$\\{\nu_{n}(k)\\}_{k\in\mathds{N}}$; these are the _switching epochs_ of
$\mathcal{Y}_{n}$. By applying Theorem 4.4 in [21]111For the values of
$\lambda_{a}$, $\lambda_{b}$ and $P$ in that theorem, take
$\lambda_{a}=\lambda_{0}$, $\lambda_{b}=\lambda_{n}-\lambda_{0}$, and $P$ as
(3.5). and recalling that $\gamma=\lambda_{0}/2$, we have that, like
$\mathcal{Y}$, the process $\mathcal{Y}_{n}$ is a uniformized chain of
$\mathcal{X}$ but at a faster rate $\lambda_{n}/2$: the transition probability
matrix of $\mathcal{Y}_{n}$ is given by

$\displaystyle P_{n}=I+(\lambda_{n}/2)^{-1}\Pi.$ (3.8)

In parallel, we construct the sequences $\\{J_{\lambda_{n}}(t)\\}_{t\geq 0}$,
$\\{\xi_{k}^{n}\\}_{k\in\mathds{N}}$ and $\\{\chi_{k}^{n}\\}_{k\in\mathds{N}}$
in the same manner as in Section 2, with $J_{\lambda_{n}}(0)=-1$. For fixed
$n$, the process $\mathcal{J}_{\lambda_{n}}$ changes state at each epoch
$\\{\xi_{k}^{n}\\}_{k\geq 1}$. Thus, for $k\geq 1$222We use the notation
$J_{\lambda_{n}}(\cdot-0)$ and $J_{\lambda_{n}}(\cdot+0)$ to denote the left
and right limits, respectively.

$\displaystyle J_{\lambda_{n}}(\chi_{k}^{n}-0)=1,\qquad
J_{\lambda_{n}}(\chi_{k}^{n}+0)=-1.$ (3.9)

To simplify our presentation, we define $S^{n}_{k}=\chi^{n}_{\nu_{n}(k)},k\geq
1;$ these denote epochs of _synchronization_ and _desynchronization_ in the
fluid approximation of $(\mathcal{B},\mathcal{B}^{*})$, for even and odd
values of $k$, respectively. To this end, we take for $t\geq 0$

$\displaystyle J^{1}_{\lambda_{n}}(t)$ $\displaystyle=J_{\lambda_{n}}(t),$
(3.10)
$\displaystyle\begin{split}J^{2}_{\lambda_{n}}(t)&=J_{\lambda_{n}}(t)\qquad\hskip
6.94072pt\mbox{for $t\in\left[S^{n}_{2k},S^{n}_{2k+1}\right)$}\\\
&=-J_{\lambda_{n}}(t)\qquad\mbox{for
$t\in\left[S^{n}_{2k+1},S^{n}_{2k+2}\right)$}\end{split}$ (3.11)

for some $k\in\mathds{N}$; see Figure 3 for an illustration. It is clear that
$\mathcal{J}^{2}_{\lambda_{n}}$ is also a Markovian jump process.

$J_{\lambda_{1}}$$J_{\lambda_{1}}^{1}$$J_{\lambda_{1}}^{2}$$0$$\chi^{1}_{1}$$S^{1}_{1}$$\chi^{1}_{3}$$\chi^{1}_{4}$$S_{2}^{1}$$\chi^{1}_{6}$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$$+$$-$${\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}-}$${\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}+}$${\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}-}$${\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}+}$${\color[rgb]{0.8,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0,0}-}$${\color[rgb]{0.8,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0,0}+}$${\color[rgb]{0.8,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0,0}-}$${\color[rgb]{0.8,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0,0}+}$${\color[rgb]{0.8,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0,0}-}$${\color[rgb]{0.8,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0,0}+}$${\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}-}$${\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}+}$${\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}-}$${\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}-}$${\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}+}$${\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}-}$${\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}+}$${\color[rgb]{0.8,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0,0}+}$${\color[rgb]{0.8,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0,0}-}$${\color[rgb]{0.8,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0,0}+}$${\color[rgb]{0.8,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0,0}-}$${\color[rgb]{0.8,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0,0}+}$${\color[rgb]{0.8,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.8,0,0}-}$${\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}-}$${\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}+}$${\color[rgb]{0,0,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.6}-}$$S_{2}^{1}$$\chi^{1}_{6}$

Figure 3: A sample path of $\mathcal{J}_{\lambda_{1}}$,
$\mathcal{J}_{\lambda_{1}}^{1}$ and $\mathcal{J}_{\lambda_{1}}^{2}$. Following
the example in Figure 2, we have
$S^{1}_{1}=\chi^{1}_{\nu_{1}(1)}=\chi^{1}_{2}$ and
$S^{1}_{2}=\chi^{1}_{\nu_{1}(2)}=\chi^{1}_{5}$.

######  Remark 3.2.

For later reference we observe that, on the one hand,
$\mathcal{J}_{\lambda_{n}}^{1}$, like $\mathcal{J}_{\lambda_{n}}$, changes
state at every epochs $\xi_{k}^{n}$. On the other hand, the dynamic behaviour
of $\mathcal{J}_{\lambda_{n}}^{2}$ is slightly different: this process
switches states at every epoch $\xi_{k}^{n}$ that lie in an open interval
$(S_{i}^{n},S_{i+1}^{n})$, but it remains continuous at the epochs
$S_{i}^{n}$: by (3.9, 3.11)

$J_{\lambda_{n}}^{2}(S_{i}^{n}-0)=J_{\lambda_{n}}^{2}(S_{i}^{n}+0)=1\quad\mbox{for
odd $i$,}\qquad
J_{\lambda_{n}}^{2}(S_{i}^{n}-0)=J_{\lambda_{n}}^{2}(S_{i}^{n}+0)=-1\quad\mbox{for
even $i$}.$

Briefly stated, synchronizing and desynchronizing epochs are recognised by
$\mathcal{J}^{1}_{\lambda_{n}}$ swicthing state while
$\mathcal{J}^{2}_{\lambda_{n}}$ stays the same.

Step 3. Finally, define
$(\mathcal{F}^{1}_{\lambda_{n}},\mathcal{F}^{2}_{\lambda_{n}})$ as

$\displaystyle
F^{i}_{\lambda_{n}}(t)=\sqrt{\lambda_{n}}\int_{0}^{t}J^{i}_{\lambda_{n}}(s)\mathrm{d}s,\qquad
t\geq 0,\ i=1,2.$ (3.12)

######  Lemma 3.3.

For each $n,k\in\mathds{N}$,

$F^{2}_{\lambda_{n}}(\chi_{k}^{n})=B^{*}(\theta_{k}^{n}).$

###### Proof.

Note that

$\displaystyle F^{2}_{\lambda_{n}}(\chi_{k}^{n})$
$\displaystyle=\sqrt{\lambda_{n}}\int_{0}^{\chi^{n}_{k}}J^{2}_{\lambda_{n}}(s)\mathrm{d}s$
$\displaystyle=\sum_{i=1}^{k}(-1)^{Y_{n}(i-1)}\left(\sqrt{\lambda_{n}}\int_{\chi^{n}_{i-1}}^{\chi^{n}_{i}}J_{\lambda_{n}}(s)\mathrm{d}s\right)$
$\displaystyle=\sum_{i=1}^{k}(-1)^{Y_{n}(i-1)}\left(F_{\lambda_{n}}(\chi_{i}^{n})-F_{\lambda_{n}}(\chi_{i-1}^{n})\right)$
$\displaystyle=\sum_{i=1}^{k}(-1)^{X(\theta_{i-1}^{n})}(B(\theta_{i}^{n})-B(\theta_{i-1}^{n}))$
$\displaystyle=B^{*}(\theta_{k}^{n}).$

∎

######  Theorem 3.4.

The process
$\bm{\mathcal{J}}_{\lambda_{n}}=(\mathcal{J}^{1}_{\lambda_{n}},\mathcal{J}^{2}_{\lambda_{n}})$
with state space $\mathcal{E}^{2}$ has the intensity matrix given by

$\displaystyle\Pi_{n}=\left[\begin{array}[]{cccc}-\lambda_{n}&0&2\alpha&\lambda_{n}-2\alpha\\\
0&-\lambda_{n}&\lambda_{n}-2\beta&2\beta\\\ 0&\lambda_{n}&-\lambda_{n}&0\\\
\lambda_{n}&0&0&-\lambda_{n}\end{array}\right].$ (3.17)

###### Proof.

For notational simplicity, we omit in this proof the dependence on
$\lambda_{n}$. By construction, the jumps in $\bm{\mathcal{J}}$ occur at the
epochs $\\{\xi_{k}^{n}\\}_{k\geq 1}$ at rate $\lambda_{n}$. Therefore, the
diagonal elements of $\Pi_{n}$ are all equal to $-\lambda_{n}$.

Consider the off-diagonal elements of the last two rows of $\Pi_{n}$. If
$J^{1}(t)=-1$, then by (3.9) the next jump occurs at some epoch
$\xi_{k^{\prime}}^{n}$ where $k^{\prime}$ is odd. Thus, $\xi_{k^{\prime}}^{n}$
cannot be one of the epochs $S_{k}^{n}$, and by Remark 3.2 both
$\mathcal{J}^{1}$ and $\mathcal{J}^{2}$ will change signs. Therefore, the only
possible transition for $\mathcal{J}$ from $(-1,-1)$ is to $(1,1)$, and from
$(-1,1)$ to $(1,-1)$.

Next, consider the first two rows of $\Pi_{n}$. If $\bm{J}(t)=(1,1)$, then
$J^{1}(t)$ and $J^{2}(t)$ are in a synchronising interval and there are two
possibilities:

  * •

with probability $(1-2\alpha/\lambda_{n})$, the next jump does not cause a
desynchronization; by Remark 3.2 the process $\mathcal{J}$ lands in $(-1,-1)$;
or

  * •

with probability $2\alpha/\lambda_{n}$, the next jump does; thus,
$\mathcal{J}^{2}$ does not change and $\mathcal{J}$ lands in $(-1,1)$.

Analogously, if $\bm{J}(t)=(1,-1)$, then $J^{1}(t)$ and $J^{2}(t)$ are in a
desynchronising interval and

  * •

with probability $(1-2\beta/\lambda_{n})$, the next jump does not cause a
synchronisation, the transition is to $(-1,1)$; or

  * •

with probability $2\beta/\lambda_{n}$, the next jump does cause a
synchronisation, and the transition is to $(-1,-1)$.

Multiplying the respective probabilities and intensities yields (3.17). ∎

######  Theorem 3.5.

With the underlying phase process
$(\mathcal{J}^{1}_{\lambda_{n}},\mathcal{J}^{2}_{\lambda_{n}})$ constructed as
in (3.10, 3.11), the bivariate flip-flop
$(\mathcal{F}^{1}_{\lambda_{n}},\mathcal{F}^{2}_{\lambda_{n}})$ converges
strongly to the process $(\mathcal{B},\mathcal{B}^{*})$.

###### Proof.

Demonstrating strong convergence of
$(\mathcal{F}^{1}_{\lambda_{n}},\mathcal{F}^{2}_{\lambda_{n}})$ to
$(\mathcal{B},\mathcal{B}^{*})$ is equivalent to proving that for all $T>0$,

$\displaystyle\lim_{n\rightarrow\infty}\sup_{0\leq s\leq
T}\left\|\left(\mathcal{F}^{1}_{\lambda_{n}}(s),\mathcal{F}^{2}_{\lambda_{n}}(s)\right)-\left(\mathcal{B}(s),\mathcal{B}^{*}(s)\right)\right\|_{\infty}=0,$
(3.18)

where $\|\cdot\|_{\infty}$ denotes the max–norm in $\mathds{R}^{2}$. Note that

$\sup_{0\leq s\leq
T}\|(\mathcal{F}^{1}_{\lambda_{n}}(s),\mathcal{F}^{2}_{\lambda_{n}}(s))-(\mathcal{B}(s),\mathcal{B}^{*}(s))\|_{\infty}\leq\sup_{0\leq
s\leq T}|\mathcal{F}^{1}_{\lambda_{n}}(s)-\mathcal{B}(s)|+\sup_{0\leq s\leq
T}|\mathcal{F}^{2}_{\lambda_{n}}(s)-\mathcal{B}^{*}(s)|,$

Then, (3.18) follows once we prove that $\mathcal{F}^{1}_{\lambda_{n}}$
strongly converges to $\mathcal{B}$, and that $\mathcal{F}^{2}_{\lambda_{n}}$
strongly converges to $\mathcal{B}^{*}$. The former is a consequence of
$\mathcal{F}^{1}_{\lambda_{n}}=\mathcal{F}_{\lambda_{n}}$ and (1.4). The
latter is an application of [21, Theorem 1.1], since $\mathcal{B}^{*}$ as
defined in (3.4) is essentially a Markov-modulated version of $\mathcal{B}$. ∎

###  3.2 Dependence of the limiting process

As previously discussed, the process $(\mathcal{B},\mathcal{B}^{*})$ happens
to be not Gaussian. In this section we examine the dependency structure of the
limiting process $(\mathcal{B},\mathcal{B}^{*})$ from Theorem 3.5. To do so,
consider the correlation coefficient function $\mathrm{Corr}(t)$ of
$\mathcal{B}$ and $\mathcal{B}^{*}$:

$\mathrm{Corr}(t)=\frac{\mathds{E}\left[B(t)B^{*}(t)\right]-\mathds{E}[B(t)]\mathds{E}[B^{*}(t)]}{\sqrt{\mbox{\emph{Var}}(B(t))}\sqrt{\mbox{\emph{Var}}(B^{*}(t))}}=\frac{\mathds{E}\left[B(t)B^{*}(t)\right]}{t}\quad\mbox{for
$t>0$.}$

######  Theorem 3.6.

The correlation coefficient function of $\mathcal{B}$ and $\mathcal{B}^{*}$ is
given by

$\displaystyle\mathrm{Corr}(t)=\frac{\alpha^{-1}-\beta^{-1}}{\alpha^{-1}+\beta^{-1}}+\frac{2\alpha(1-e^{-t(\alpha+\beta)})}{t(\alpha+\beta)^{2}}.$
(3.19)

###### Proof.

Let $e_{q}$ be an exponential random variable of rate $q$, independent of
$\mathcal{B}$ and $\mathcal{X}$. We compute

$\mathds{E}\left[B(e_{q})B^{*}(e_{q})\right]=q\int_{0}^{\infty}e^{-qt}\mathds{E}\left[B(t)B^{*}(t)\right]\mathrm{d}t,$

and perform Laplace inversion to obtain $\mathds{E}\left[B(t)B^{*}(t)\right]$.
To that end, consider the arrival epochs $\\{\theta^{0}_{k}\\}_{k\geq 1}$ of
the Poisson process $\mathcal{N}_{\lambda_{0}}$ with rate
$\lambda_{0}=2\gamma=2(\alpha+\beta)$. Since $\theta^{0}_{k}\rightarrow\infty$
as $k\rightarrow\infty$, we have

$\displaystyle\lim_{k\rightarrow\infty}B(\theta^{0}_{k}\wedge
e_{q})B^{*}(\theta^{0}_{k}\wedge e_{q})=B(e_{q})B^{*}(e_{q}).$

Furthermore, for all $k\geq 1$,

$|B(\theta^{0}_{k}\wedge e_{q})B^{*}(\theta^{0}_{k}\wedge
e_{q})|\leq\left(\sup_{s\leq e_{q}}|B(s)|\right)\left(\sup_{s\leq
e_{q}}|B^{*}(s)|\right).$

Consequently, Lemma 3.7 below and the Dominated Convergence Theorem imply that

$\displaystyle\mathds{E}\left[B(e_{q})B^{*}(e_{q})\right]$
$\displaystyle=\lim_{k\rightarrow\infty}\mathds{E}\left[B(\theta^{0}_{k}\wedge
e_{q})B^{*}(\theta^{0}_{k}\wedge e_{q})\right].$ (3.20)

Next, for $k\geq 1$,

$\displaystyle\mathds{E}[B(\theta^{0}_{k}\wedge
e_{q})B^{*}(\theta^{0}_{k}\wedge e_{q})]$
$\displaystyle=\mathds{E}\left[\left(\sum_{i=1}^{k}B(\theta^{0}_{i}\wedge
e_{q})-B(\theta^{0}_{i-1}\wedge
e_{q})\right)\left(\sum_{i=1}^{k}B^{*}(\theta^{0}_{i}\wedge
e_{q})-B^{*}(\theta^{0}_{i-1}\wedge e_{q})\right)\right]$
$\displaystyle=\mathds{E}\left[\left(\sum_{i=1}^{k}B(\theta^{0}_{i}\wedge
e_{q})-B(\theta^{0}_{i-1}\wedge
e_{q})\right)\left(\sum_{i=1}^{k}(-1)^{Y(i-1)}\left(B(\theta^{0}_{i}\wedge
e_{q})-B(\theta^{0}_{i-1}\wedge e_{q})\right)\right)\right]$
$\displaystyle=\mathds{E}\left[\sum_{i=1}^{k}(-1)^{Y(i-1)}\left(B(\theta^{0}_{i}\wedge
e_{q})-B(\theta^{0}_{i-1}\wedge e_{q})\right)^{2}\right]$
$\displaystyle=\mathds{E}\left[\mathds{E}\left[\left.\sum_{i=1}^{k}(-1)^{Y(i-1)}\left(B(\theta^{0}_{i}\wedge
e_{q})-B(\theta^{0}_{i-1}\wedge e_{q})\right)^{2}\;\right|\;\\{Y(k)\\}_{k\geq
0}\right]\right]$
$\displaystyle=\mathds{E}\left[\sum_{i=1}^{k}(-1)^{Y(i-1)}\mathds{E}\left[(B(\theta^{0}_{i}\wedge
e_{q})-B(\theta^{0}_{i-1}\wedge e_{q}))^{2}\right]\right]$
$\displaystyle=\mathds{E}\left[\sum_{i=1}^{k}(-1)^{Y(i-1)}\mathds{E}\left[\mathds{E}\left[(B(\theta^{0}_{i}\wedge
e_{q})-B(\theta^{0}_{i-1}\wedge
e_{q}))^{2}\;\;|\;\;\theta^{0}_{i},\theta^{0}_{i-1},e_{q}\right]\right]\right]$
$\displaystyle=\mathds{E}\left[\sum_{i=1}^{k}(-1)^{Y(i-1)}\mathds{E}\left[\theta^{0}_{i}\wedge
e_{q}-\theta^{0}_{i-1}\wedge e_{q}\right]\right]$
$\displaystyle=\mathds{E}\left[\sum_{i=1}^{k}(-1)^{Y(i-1)}\frac{\gamma^{i-1}}{(\gamma+q)^{i}}\right]\quad\mbox{since
$\theta_{k}^{0}\sim\mbox{Erlang}(k,\gamma)$},$
$\displaystyle=\sum_{i=1}^{k}\left[\mathds{P}(Y(i-1)=0)-\mathds{P}(Y(i-1)=1)\right]\frac{\gamma^{i-1}}{(\gamma+q)^{i}}$
$\displaystyle=\frac{1}{\gamma+q}\sum_{i=0}^{k-1}[\mathds{P}(Y(i)=0)-\mathds{P}(Y(i)=1)]\left(\frac{\gamma}{\gamma+q}\right)^{i}.$

By (3.5), for $m\geq 1$ the $m$th-step transition probability matrix of
$\mathcal{Y}$ is given by

$\displaystyle(P_{n})^{m}=\left[\begin{array}[]{cc}\beta/\gamma&\alpha/\gamma\\\
\beta/\gamma&\alpha/\gamma\end{array}\right],$

which implies that, independently of the distribution of $Y(0)$,
$\mathds{P}(Y(m)=0)=\beta/\gamma$ and $\mathds{P}(Y(m)=1)=\alpha/\gamma,$ for
$m\geq 1.$ This, together with the fact that by construction $P(Y(0)=0)=1$ and
$P(Y(0)=1)=0$, gives

$\displaystyle\mathds{E}\left[B(\theta^{0}_{k}\wedge
e_{q})B^{*}(\theta^{0}_{k}\wedge e_{q})\right]$
$\displaystyle=\left[\frac{1}{\gamma+q}\left(\frac{\beta-\alpha}{\gamma}\right)\sum_{i=0}^{k-1}\left(\frac{\gamma}{\gamma+q}\right)^{i}\right]+\frac{2\alpha}{\gamma(\gamma+q)}.$
(3.21)

Substituting (3.21) into (3.20) leads to

$\displaystyle\mathds{E}\left[B(e_{q})B^{*}(e_{q})\right]$
$\displaystyle=\frac{\beta-\alpha}{\gamma
q}+\frac{2\alpha}{\gamma(\gamma+q)}.$

Applying Laplace inversion to $q^{-1}\mathds{E}[B(e_{q})B^{*}(e_{q})]$ yields

$\mathds{E}\left[B(t)B^{*}(t)\right]=\frac{(\beta-\alpha)t}{\alpha+\beta}+\frac{2\alpha(1-e^{-t(\alpha+\beta)})}{(\alpha+\beta)^{2}}=\frac{(\alpha^{-1}-\beta^{-1})t}{\alpha^{-1}+\beta^{-1}}+\frac{2\alpha(1-e^{-t(\alpha+\beta)})}{(\alpha+\beta)^{2}},$

and thus (3.19) follows. ∎

######  Lemma 3.7.

Let $e_{q}$ be an exponential random variable of rate $q$, independent of
$\mathcal{B}$ and $\mathcal{X}$. We have

$\displaystyle\mathds{E}\left[\left(\sup_{s\leq
e_{q}}|B(s)|\right)\left(\sup_{s\leq e_{q}}|B^{*}(s)|\right)\right]{<\infty}.$
(3.22)

###### Proof.

By the Cauchy-Schwarz inequality, we have

$\displaystyle\mathds{E}\left[\left(\sup_{s\leq
e_{q}}|B(s)|\right)\left(\sup_{s\leq
e_{q}}|B^{*}(s)|\right)\right]\leq\left\\{\mathds{E}\left[\left(\sup_{s\leq
e_{q}}|B(s)|\right)^{2}\right]\mathds{E}\left[\left(\sup_{s\leq
e_{q}}|B^{*}(s)|\right)^{2}\right]\right\\}^{1/2}$
$\displaystyle=\mathds{E}\left[\left(\sup_{s\leq
e_{q}}|B(s)|\right)^{2}\right]\qquad\mbox{since $\mathcal{B}$ and
$\mathcal{B}^{*}$ have the same distribution,}$
$\displaystyle\leq\mathds{E}\left[\left(\sup_{s\leq e_{q}}B(s)-\inf_{s\leq
e_{q}}B(s)\right)^{2}\right]$ $\displaystyle=\mathds{E}\left[\left(\sup_{s\leq
e_{q}}B(s)\right)^{2}\right]+\mathds{E}\left[\left(-\inf_{s\leq
e_{q}}B(s)\right)^{2}\right]+2\mathds{E}\left[\left(\sup_{s\leq
e_{q}}B(s)\right)\left(-\inf_{s\leq e_{q}}B(s)\right)\right]$
$\displaystyle\leq\mathds{E}\left[\left(\sup_{s\leq
e_{q}}B(s)\right)^{2}\right]+\mathds{E}\left[\left(-\inf_{s\leq
e_{q}}B(s)\right)^{2}\right]+2\left\\{\mathds{E}\left[\left(\sup_{s\leq
e_{q}}B(s)\right)^{2}\right]\mathds{E}\left[\left(-\inf_{s\leq
e_{q}}B(s)\right)^{2}\right]\right\\}^{1/2}.$

By Wiener-Hopf factorisation, $\sup_{s\leq e_{q}}B(s)$ and $-\inf_{s\leq
e_{q}}B(s)$ are exponentially distributed. Thus, each expectation in the final
sum is finite and (3.22) follows. ∎

Elementary but tedious machinery was used in the proof of Theorem 3.6. A more
elegant approach is by considering a more general setting, which will be done
in Section 4.

######  Remark 3.8.

Theorem 3.6 implies that the limiting process $(\mathcal{B},\mathcal{B}^{*})$
has a time-dependent correlation function, which starts in $1$, is strictly
decreasing and converges to
$(\alpha^{-1}-\beta^{-1})/(\alpha^{-1}+\beta^{-1})$ as $t\rightarrow\infty$.
From a modelling perspective, this provides an alternative to classic
correlated bivariate Brownian motion models, for which the correlation
function remains constant over time.

A completely analogous construction can be made so that $\mathcal{B}^{*}$
starts in a desynchronized environment, that is, $\mathds{P}(Y(0)=1)=1$. A
slight modification to the proof of Theorem 3.6 shows that the correlation
coefficient for such a construction is given by

$\mathrm{Corr}(t)=\frac{\alpha^{-1}-\beta^{-1}}{\alpha^{-1}+\beta^{-1}}-\frac{2\beta(1-e^{-t(\alpha+\beta)})}{t(\alpha+\beta)^{2}}.$

##  4 Reflection at MAP times

In this section, we construct a flip-flop approximation to a new class of
bivariate Brownian motion processes with a more flexible time-dependent
correlation function, which we call _two-dimensional MAP alternating Brownian
motion_. They are a generalisation of the two-dimensional exponentially
alternating Brownian motion, as the synchronisation and desynchronisation
epochs now occur according to the arrival epochs of a continuous-time MAP. In
continuous time, a MAP of parameters $(\bm{b},C,D)$ is a counting process
$\mathcal{K}$ driven by an underlying Markov jump process with initial
distribution $\bm{b}$ and intensity matrix $C+D$. The epochs of increase of
$\mathcal{K}=\\{K(t)\\}_{t\geq 0}$ coincide with those jumps epochs which
occur due to the intensities of the matrix $D$. Analogously, a discrete-time
MAP is driven by a Markov chain instead of a Markov jump process. See [19] for
a detailed exposition on the topic.

######  Definition 4.1.

Let $\mathcal{K}=\\{K(t)\\}_{t\geq 0}$ be a continuous-time MAP$(\bm{b},C,D)$,
and $\mathcal{B}$ be a standard Brownian motion. Define
$\mathcal{B}^{*}=\\{B^{*}(t)\\}_{t\geq 0}$ such that

$B^{*}(t)=\int_{0}^{t}(-1)^{K(s)}\mathrm{d}B_{s},\quad t\geq 0.$

The process $(\mathcal{B},\mathcal{B}^{*})$ is called a two-dimensional MAP
alternating Brownian motion.

###  4.1 Construction

Let $(\Omega,\mathscr{F},\mathds{P})$ be a probability space as in Section 3,
where the elements $\mathcal{B}$, $\mathcal{N}_{\lambda_{n}}$,
$\mathcal{J}_{\lambda_{n}}$, $\mathcal{F}_{\lambda_{n}}$,
$\\{\theta^{n}_{k}\\}_{k\geq 0}$, $\\{\chi^{n}_{k}\\}_{k\geq 0}$, $n\geq 0$,
are defined in Section 3.

Step 1. We start by constructing the MAP $\mathcal{K}$. W.l.o.g. suppose that

$\gamma=\lambda_{0}/2\geq\max_{i\in\mathcal{S}}|C_{ii}|.$

We enlarge the space $(\Omega,\mathscr{F},\mathds{P})$ so that it supports a
discrete-time MAP $\mathcal{M}=\\{M(k)\\}_{k\geq 0}$ with parameters
$(\bm{b},I+\gamma^{-1}C,\gamma^{-1}D)$ and the underlying Markov chain
$\mathcal{Y}=\\{Y(k)\\}_{k\geq 0}$. Define $\mathcal{K}=\\{K(t)\\}_{t\geq 0}$
and $\mathcal{X}=\\{X(t)\\}$ by

${K(t)}=M(N_{\lambda_{0}}(t)),\qquad{X(t)}=Y(N_{\lambda_{0}}(t)),\quad t\geq
0.$

Reversed uniformization implies that $\mathcal{K}$ is a continuous-time
$\mbox{MAP}(\bm{b},C,D)$ with the underlying Markov jump process
$\mathcal{U}$.

Analogous to the construction in Section 3.1 and in a slight abuse of
notation, we define here a new sequence $\\{\ell(k)\\}_{k\geq 0}$. Previously,
in the exponentially alternating BM, the jump epochs of Markov jump process
$\mathcal{X}$ mark the synchronisation/desynchronisation epochs; the $k$th
jump of $\mathcal{X}$ occurs at $\theta^{0}_{\ell(k)}$, i.e. the $\ell(k)$th
arrival of $\mathcal{N}_{\lambda_{0}}$, where $\\{\ell(k)\\}_{k\geq 1}$ are
the epochs at which the uniformized chain $\mathcal{Y}$ of $\mathcal{X}$
changes states. Here, in the MAP alternating BM, the jump epochs of the MAP
$\mathcal{K}$ mark the synchronisation/desynchronisation epochs; the $k$th
jump of $\mathcal{K}$ occurs at $\theta^{0}_{\ell(k)}$, where
$\\{\ell(k)\\}_{k\geq 0}$ are the arrivals epochs of the discrete-time MAP
$\mathcal{M}$:

$\ell(k)=\inf\\{\nu\geq 0:M(\nu)=k\\},$

for $k\geq 0$. Note that the process $\mathcal{U}$ may or may not jump at each
epoch $\theta_{k}^{0}$.

Step 2. Here, we construct the phase process
$(\mathcal{J}_{\lambda_{n}}^{1},\mathcal{J}_{\lambda_{n}}^{2})$. First, for
$n\geq 1$ we define the processes $\mathcal{M}^{n}=\\{M^{n}(k)\\}_{k\geq 0}$
and $\mathcal{V}^{n}=\\{V^{n}(k)\\}$ by

$M^{n}(k)=K(\theta^{n}_{k}),\qquad V^{n}(k)=U(\theta^{n}_{k}).$

By [21, Theorem 4.4], $\mathcal{M}^{n}$ is a discrete-time
MAP$(\bm{b},I+(\lambda_{n}/2)^{-1}C,(\lambda_{n}/2)^{-1}D)$ with the
underlying Markov chain $\mathcal{V}^{n}$. We do not make direct use of the
process $\mathcal{M}^{n}$, but we do use $\mathcal{V}^{n}$ and define the
processes $\mathcal{U}^{n}=\\{U^{n}(t)\\}_{t\geq 0}$ as follows:

$\displaystyle U^{n}(t)=V^{n}(j)\quad\mbox{if
$t\in[\chi^{n}_{j},\chi^{n}_{j+1})$ for some $j\geq 0$.}$ (4.1)

The processes $\mathcal{U}^{n}$ serve as approximations for the process
$\mathcal{U}$.

Since by construction the processes $\\{\mathcal{N}_{\lambda_{n}}\\}_{n\geq
0}$ are nested, for each $n,k\geq 0$ there exists a _unique_
$\nu_{n}(k)\in\mathds{N}$ such that
$\theta_{(\nu_{n}(k))}^{n}=\theta_{\ell(k)}^{0}$, so that the $\nu_{n}(k)$-th
arrival epoch of $\mathcal{N}_{\lambda_{n}}$ coincides with the jump of
$\mathcal{K}$ from $m-1$ to $m$; it is a synchronisation point if $m$ is even,
and a desynchronisation point if $m$ is odd.

Finally, we define $S^{n}_{k}=\chi^{n}_{\nu_{n}(k)}$ as in Section 3 and

$\displaystyle J^{1}_{\lambda_{n}}(t)$
$\displaystyle=(J_{\lambda_{n}}(t),U^{n}(t))\quad\mbox{for all}\quad t\geq 0$
(4.2)
$\displaystyle\begin{split}J^{2}_{\lambda_{n}}(t)&=(J_{\lambda_{n}}(t),U^{n}(t))\quad\mbox{for
}t\in\left[S^{n}_{2k},S^{n}_{2k+1}\right)\\\
&=(-J_{\lambda_{n}}(t),U^{n}(t))\quad\mbox{ for
}t\in\left[S^{n}_{2k+1},S^{n}_{2k+2}\right)\end{split}$ (4.3)

for some $k\in\mathds{N}$.

Step 3. Finally, define
$(\mathcal{F}^{1}_{\lambda_{n}},\mathcal{F}^{2}_{\lambda_{n}})$ where

$\displaystyle
F^{i}_{\lambda_{n}}(t)=\sqrt{\lambda_{n}}\int_{0}^{t}\pi_{1}(J^{i}_{\lambda_{n}}(s))\mathrm{d}s,\quad
t\geq 0,i\in\\{1,2\\},$ (4.4)

with $\pi_{1}:\\{1,-1\\}\times\mathcal{S}\mapsto\\{1,-1\\}$ denoting the first
coordinate projection.

######  Lemma 4.2.

For each $n,k\geq 0$,

$F^{2}_{\lambda_{n}}(\chi_{k}^{n})=B^{*}(\theta_{k}^{n}).$

The proof of this lemma is analogous to Lemma 3.3, and is omitted.

######  Theorem 4.3.

The phase process
$\bm{\mathcal{J}}_{\lambda_{n}}=(\mathcal{J}^{1}_{\lambda_{n}},\mathcal{J}^{2}_{\lambda_{n}})$,
which lives on space $\mathcal{E}^{2}\times\mathcal{S}$, has the initial
distribution $(\bm{0},\bm{0},\bm{0},\bm{b})$ and the intensity matrix

$\displaystyle
Q_{n}=\left[\begin{array}[]{cccc}-\lambda_{n}I&0&2D&\lambda_{n}I+2C\\\
0&-\lambda_{n}I&\lambda_{n}I+2C&2D\\\ 0&\lambda_{n}I&-\lambda_{n}I&0\\\
\lambda_{n}I&0&0&-\lambda_{n}I\end{array}\right],$ (4.9)

where $I$ is an $|\mathcal{S}|\times|\mathcal{S}|$ identity matrix.

###### Proof.

Like in the proof of Theorem 4.3, the process
$(\mathcal{J}^{1}_{\lambda_{n}},\mathcal{J}^{2}_{\lambda_{n}})$ changes states
at rate $\lambda_{n}$. Furthermore, Remark 3.2 remains relevant: the first
coordinate of $\mathcal{J}^{1}_{\lambda_{n}}$ switches state at each epoch
$\\{\xi_{k}^{n}\\}_{k\geq 1}$, while the first coordinate of
$\mathcal{J}^{2}_{\lambda_{n}}$ does so only in open intervals
$\left(S_{i}^{n},S_{i+1}^{n}\right)$, and remains constant at the epochs
$S_{i}^{n}$. In addition, by (4.1), $U^{n}(t)$ remains constant when
$\mathcal{J}^{1}_{\lambda_{n}}$ jumps from $-1$ to $+1$, at the epochs
$\\{\xi_{2k+1}^{n}\\}_{k\geq 0}$.

Therefore,

  * •

if $\bm{J}_{\lambda_{n}}(t)=(-1,-1,j)$ for some $j\in\mathcal{S}$, then the
next jump is necessarily to $(1,1,j)$; similarly, if
$\bm{J}_{\lambda_{n}}(t)=(-1,1,j)$, the next jump is necessarily to
$(1,-1,j)$. This justifies the structure of the last 2 blocks of rows in
(4.9).

  * •

If $\bm{J}_{\lambda_{n}}(t)$ is in $\\{(1,1)\\}\times\mathcal{S}$, then there
are two possibilities:

    * –

either the associated Markov chain $\mathcal{V}^{n}$ does not induce an
arrival in the MAP process $\mathcal{M}^{n}$, no desynchronization happens and
$\bm{\mathcal{J}}_{\lambda_{n}}$ lands in some state in
$\\{(-1,-1)\\}\times\mathcal{S}$ according to the transition matrix
$I+(\lambda_{n}/2)^{-1}C$

    * –

or $\mathcal{V}^{n}$ induces an arrival for the MAP process $\mathcal{M}^{n}$,
a desynchronization event occurs, and $\bm{\mathcal{J}}_{\lambda_{n}}$ lands
in some state in $\\{(-1,1)\\}\times\mathcal{S}$ according to the transition
matrix $(\lambda_{n}/2)^{-1}D$,

  * •

If $\bm{J}_{\lambda_{n}}(t)$ is in $\\{(1,-1)\\}\times\mathcal{S}$, then there
are two possibilities:

    * –

either $\mathcal{V}^{n}$ induces a synchronization event at its next
transition, with transition matrix $(\lambda_{n}/2)^{-1}D,$ and the new phase
is in $\\{(-1,-1)\\}\times\mathcal{S}$,

    * –

or the next transition does not induce a new synchronization event, the
transition matrix here is $I+(\lambda_{n}/2)^{-1}C$, and the new state is in
$\\{(-1,-1)\\}\times\mathcal{S}$.

Multiplying the probabilities accordingly yields (4.9). ∎

######  Theorem 4.4.

The bivariate flip-flop
$(\mathcal{F}^{1}_{\lambda},\mathcal{F}^{2}_{\lambda})$ with the underlying
process $(\mathcal{J}^{1}_{\lambda},\mathcal{J}^{2}_{\lambda})$ converges
strongly to $(\mathcal{B},\mathcal{B}^{*})$ as $\lambda\rightarrow\infty$.

This proof is similar to that of Theorem 3.5, and is omitted.

###  4.2 Dependence of the limiting process

Here, we assess the correlation between $\mathcal{B}$ and $\mathcal{B}^{*}$.
Before doing so, we recall that a phase-type distribution of parameters
$(\bm{b},T)$ corresponds to the distribution of the absorption time of a
Markov jump process with initial distribution $\bm{b}$ and subintensity matrix
$T$. The mean of such a distribution is given by $\bm{b}(-T^{-1})\bm{e}$,
where $\bm{e}$ denotes a column vector of $1$s of appropriate size. See [19]
for more details.

######  Theorem 4.5.

For any $q>0$,

$\displaystyle\int_{0}^{\infty}e^{qt}\mathds{E}\left[B(t)B^{*}(t)\right]\mathrm{d}t=-\frac{1}{q}\bm{b}[I+(C-qI)^{-1}D][C-qI-D(C-qI)^{-1}D]^{-1}\bm{e}.$
(4.10)

###### Proof.

Let $e_{q}$ be an independent exponential random variable with rate $q$. Then,

$\displaystyle\mathds{E}\left[\mathcal{B}(e_{q})\mathcal{B}^{*}(e_{q}\right)]$
$\displaystyle=\mathds{E}\left[\left(\int_{0}^{e_{q}}\mathrm{d}B_{s}\right)\left(\int_{0}^{e_{q}}(-1)^{K(s)}\mathrm{d}B_{s}\right)\right]$
$\displaystyle=\mathds{E}\left[\int_{0}^{e_{q}}\mathds{1}_{\\{K(s)\in
2\mathds{N}\\}}\mathrm{d}s\right]-\mathds{E}\left[\int_{0}^{e_{q}}\mathds{1}_{\\{K(s)\in
2\mathds{N}+1\\}}\mathrm{d}s\right],$ (4.11)

where $\mathds{1}_{\\{\cdot\\}}$ is the indicator function. The first summand
in (4.11) corresponds to the expected occupation time of the process
$\\{K(t)\\}_{t\geq 0}$ in $2\mathds{N}$ up to $e_{q}$. This total lifetime has
a phase-type distribution with parameters $(\bm{b},(C-qI)+D(-C+qI)^{-1}D)$:

  * •

the term $(C-qI)$ corresponds to the original transition intensities on the
event they happen before $e_{q}$,

  * •

the term $D(-C+qI)^{-1}D$ corresponds to the transitions arising from the end
of an even sojourn time into the start of the following even sojourn time, in
the case $e_{q}$ does not happen between such epochs.

In a similar way, we can argue that the second summand in (4.11) corresponds
to the expected value of a phase-type-distributed random variable of
parameters $(\bm{b}(-C+qI)^{-1}D,(C-qI)+D(-C+qI)^{-1}D)$. Thus, Theorem 4.5
follows. ∎

######  Remark 4.6.

We can recover the formula in Theorem 3.6 by letting

$\displaystyle\bm{b}=(1,0),\quad C=\left[\begin{array}[]{rr}-\alpha&0\\\
0&-\beta\end{array}\right],\quad D=\left[\begin{array}[]{rr}0&\alpha\\\
\beta&0\end{array}\right],$

and performing Laplace inversion on (4.10). This would give us a bivariate
flip-flop with 8 states, but they are partitioned into two communication
classes of size 4, one of which is never visited. Thus it can be reduced to
the original 4-state model.

## Acknowledgements.

The second and third authors are affiliated with Australian Research Council
(ARC) Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS).
We would like to also acknowledge the support of the ARC DP180103106 grant.

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