# Large deviations for spatial telecommunication systems: The boolean mode

By A. K. Boahen1, T. Katsekpor2 and K. Doku-Amponsah3

1AIMS Ghana

2Department of Mathematics, University of Ghana, Legon,Accra

3Department of Statistics and Actuarial Science, University of Ghana,
Legon,Accra

3 Email<EMAIL_ADDRESS>
> Abstract. Spatial telecommunication systems have evolved along the years,
> leading to some concerns that telecommunication companies are facing today.
> The main inquietude is the ability to provide quality service to customers
> or users in a dense regime. Therefore, questions such as : what is the best
> possible configurations of base stations and users that maximizes quality
> service? Is it possible to estimate and control the probability of bad
> service, which may be seen as a rare event? and many more arise. These
> questions often involve estimating the tail distribution of events, which
> falls under the scope of large deviation principles. In this article, we
> associate with the Boolean model, the empirical marked measure which will
> serve as a statistic for the intensity measure of the Marked Poisson Point
> Process of devices or users and the empirical connectivity measure which
> will serve as a statistic for coverage probability density of the spatial
> telecommunication area. For these empirical measures, prove large deviation
> principle (LDP) for well-defined empirical measures.

Keywords: Boolean Model, Marked Poisson Point Processes, Empirical Measures,
Large deviations, Coverage probability density, Intensity measure, Relative
Entropy

AMS Subject Classification: 60F10, 05C80, 68Q87

##  1\. Introduction

Throughout the ages, telecommunication has played a very important role in
humanity. It has become an integral part of our world. From sending a text
message or an email, a WhatsApp video or voice note, or even making a
telephone call, one cannot do without any means of communication in general
and telecommunication, in particular. While telecommunication took a handful
number of forms at the beginning, it has grown at a mind-blowing rate and
presents itself under various forms. For example, smoke signals were used in
some part of Northern America and India to communicate whereas talking drums
was a means of communication in some parts of Africa, with notable example
being the Akan tribe in Ghana[15][14]. Telecommunication has evolved in ways
that no one would have thought of some centuries ago.

The evolution of telecommunication systems has made it more accessible and
hence caused a remarkable growth in the number of telecommunications users.
The increase in number of users sometimes creates perturbations in the system,
which may lead to connectivity issues and poor service. The settings of
telecommunications provide mathematicians with a very rich structure. Many
researchers have shown interest in the mathematical aspect of
telecommunication systems. It has been observed that the spatial configuration
of users in a telecommunication area is a very important aspect of the
systems. A tool that has been developed and has gained a lot of attention
among those whose research interest is in mathematical modelling of
telecommunication is stochastic geometry. The area of stochastic geometry has
provided researchers with important concept and techniques to study spatial
telecommunications.

In modelling a telecommunication system, we are mostly concerned with
questions of connectivity and coverage area. On one hand, the subject of
coverage consists of knowing the extent to which the signal emanating from
users can cover the communication area while on the other hand, connectivity
addresses the issue of the distance a message can reach in the communication
area. Trying to keep pace with the new forms of telecommunications, the most
notable being wireless networks, researchers are always trying to find the
optimal way in which users of these systems can communicate. In some settings,
the message is sent from one user to the base station, and then relayed from
the station to the user it is directed to. In some other settings (which has
been adopted in recent years), authors considered messages that travel
according to a multihop-functionality, that is from one user to the other
until it reaches its destination instead of travelling to a base station
before being sent back to the user the message has been destined to. According
to [4], the multihop-functionality mode of communication is the main idea
behind next-generation wireless networks. The precondition in such networks is
the ability to provide, on average, quality service to users. For such events
where the service quality deteriorates, there is a need to be able to control
and estimate the probability of occurrence [11]. The mathematical theory used
to accomplish such a task is known as large deviations’ theory.

Large deviations’ theory may be seen as a collection of mathematical
techniques or tools of a stochastic nature, that are employed in the
description or estimation of the asymptotic behavior of extremely rare events.
The theory of large deviation finds its applications in many areas, of which
the conventional ones are risk management and information theory.

Doku-Amponsah, in [6], defined empirical neighborhood and pair measures for a
finite colored graph. He established the large deviation principle (LDP) for
these two measures for a class of colored random graphs and their joint large
deviation principle using the technique of change of measure, the method of
types and the method of mixtures. He then proceeded to derive the LDP for the
degree distribution of the famous Erdös-Renyi graphs in the sparse case.
Finally, with the established LDP for these measures, Doku-Amponsah
demonstrated the asymptotic equipartition properties for hierarchical and
networked structures.

Hirsch et al., in 2016, proved an LDP for the empirical measure of connectable
receivers in a wireless network. They associated to individual transmitters a
set of connectable receivers with SINR larger than a fixed threshold. The
authors identified the receivers connectable to the origin of the Euclidean
space used in modelling the communication area as a Cox point process and
proceeded to establish the LDP for the rescaled process of the receivers as
the connectivity threshold was approaching zero. In addition to the technical
subtleties of their work, the authors were able to reconcile their findings
with importance sampling by decreasing the discrepancy in quantifying the
probability of rare events. The paper used the Dawson-Gärtner technique and
the contraction principle to achieve its purpose.[11].

Umar et.al in [21], studied the joint LDP of the empirical spin and bond
measures of the uniformly random $d$-regular graph. He assigned spin values to
the vertices of the graph using the Potts model. The author then used the
method of types to derive the rate function of the joint LDP. He noticed that
the rate function obtained was a sum of two entropies and the typical behavior
of the graph was seen when the rate function was equal to zero. The condition
for obtaining the typical behavior was achieved when the empirical measure was
the same as the product measure of the spin empirical measure and the bond
measure.

In [10], the authors, derived ”the large deviation principle for the space-
time evolution of frustrated transmitters” when the number of transmitters
becomes large. The transmitters and relays were set up in a bounded region of
$\mathbb{R}^{d},d\in\mathbb{N}$, which represented the communication area. The
number and position of transmitters were random, whereas the relays were
fixed. In their dissertation, a frustrated transmitter is one who selects an
occupied relay and using exponential approximation techniques, the authors
proved that the family of measure-valued processes representing the proportion
of frustrated transmitters satisfied an LDP with good rate function given by
the infimum of an entropy.

Enoch Sakyi-Yeboah et al, in the paper titled ”Large deviation principle,
Sharron-McMillan-Brieman theorem for super-critical telecommunication
networks”, obtained a large deviation asymptotics for super-critical
communication networks modelled as a Signal-to-Interference-Noise-Ratio (SINR)
network. The authors proved on the scales $\lambda$ and
$\lambda^{2}a_{\lambda}$, the LDP for two well-defined measures,the power and
the empirical connectivity measures, where $\lambda$ was the intensity of a
poisson point process (P.P.P) which was used to define the SINR random
network. They proved with ingeniosity the asymptotic equipartition property
and the local large deviation principle (LLDP) of the SINR networks and
established a classical McMillan theorem for the network. The authors
discovered a bound on the cardinality of the space of the SINR networks when
the probability of connectivity of the network satisfied some convergence
assumptions. See, [19].

Over the last few years, there has been a keen interest from researchers in
studying the optimal way to route messages in a Telecommunication network
under the ’famous’ Gibbs measure. Most literatures in the field (to the best
of my knowledge) base on the well-known SINR (Signal-to-Interference-Noise-
Ratio) to derive the Large deviation principles of some well-defined measures,
which will therefore help in establishing the Gibbsian structure of the system
at hand.

In this work, the motivation for deriving the joint LDP for the empirical
connectivity and marked measures in a telecommunication system is to be able
to establish the Gibbsian structure of the system. In fact, by deriving the
rate function of the joint large deviation of the empirical pair and marked
measures, the typical behavior of the spatial telecommunication system will be
detected. It should be noted that our two measures completely described our
model used for the telecommunication system. Therefore, via the rate function,
which was expressed as an entropy function, the Gibbs distribution will be
derived and the interacting effects of the systems evaluated. The different
values of the two measures represent various configurations of users in the
communication area and be seen as various thermodynamics states under the
Gibbs measure. This research then provide an avenue to find the best
configuration of the communication area under the Boolean model that maximizes
the service quality and yield the least communication breach.

Here, we are interested in deriving the joint large deviation principle for
the empirical marked and pair measure in a telecommunication system modelled
as Boolean network. Specifically, we will estimate the rate function of the
LDP for the empirical marked measure, the conditional pair measure given the
empirical marked measure and the joint empirical marked and pair measure in
terms of relative entropies.

The subsequent parts of this article are structured in the following
arrangements. In Section 2, we provide some mathematical preliminaries
necessary for the understanding of the work.Thus, presents the Boolean model
and the methods used to establish the large deviation principles of the
empirical measures defined in this literature. Further, in Section 2, we also
calculate the probability of connection between two users, and we derive the
necessary assumptions for convergence. The main results of the article is also
presented in Section 2. In Section 3 we prove state and proof LDP for the
empirical marked measure. section 4 contain LDP for the empirical connectivity
measure conditional on a given empirical mark measure. Section 5 which is the
last Section, draws the conclusions based on the discoveries in the work and
identify future research avenues.

##  2\. MAIN RESULTS

###  2.1. The Boolean Model for Telecommunication Systems

Fix a dimension $d\in\mathbb{N}$ and a measureable set
$D\subset\mathbb{R}^{d}$ with respect to the Borel-Sigma algebra
$B(\mathbb{R}^{d}).$ Denote by $Leb$ the Lebesgues measure on
$\mathbb{R}^{d}.$ Let $\mu_{\lambda}$ be positive measure on $D$ with
$\mu_{\lambda}(D)=1$.Given an intensity measure, $\mu_{\lambda}:D\to[0,1]$, a
coverage probability density $Q_{\lambda}$ from $D$ to $\mathbb{R}_{+}$, we
define the Boolean network as follows:

  * •

We pick $\mathbb{X}=(X_{i})_{i\in I}$ a Poisson Point Process (PPP) with
intensity measure $\mu_{\lambda}:D\to[0,\infty),$ representing the
configuration of devices or users in a communication space
$D\subseteq\mathbb{R}^{d}$.

  * •

Given $X,$ we assign each $X_{i}$ a random coverage area $B_{X_{i}}=B_{i}$
independently according to $Q_{\lambda}$ the probability distribution of the
volume of the balls centered in $D$.

  * •

For any two marked points $((X_{i},B_{i}),(X_{j},B_{j}))$ we connect an edge
iff $B_{i}\cap B_{j}\not=\emptyset$.

We consider $X^{o}(\mathbb{X},Q_{\lambda})=\Big{\\{}\Big{[}(X_{i},B_{i}),i\in
I\Big{]},\,E\Big{\\}}$ under the joint law of the marked P.P.P. and the
network. We shall interpret $X^{o}$ as a Boolean random network and
$X_{i}^{o}:=(X_{i},B_{i})$ the coverage area of device $i.$ We assume that
there is a real sequence $a_{\lambda}$ and a functions ${\mathcal{C}}$, $\mu$
such that

$a_{\lambda}\mu_{\lambda}(x)\to\mu(x)\,\,\mbox{
and}\,\,a_{\lambda}^{-1}Q_{\lambda}(r|x)\to Q(r|x),$

where $\lambda a_{\lambda}\to 1$ or $\lambda a_{\lambda}\to 0$ or $\lambda
a_{\lambda}\to\infty.$ We shall restrict to the case where $\lambda
a_{\lambda}\to 1$. Thus, we shall focus on the critical Boolen Random
Networks. We write $\displaystyle{\mathcal{B}}=\Big{\\{}B(x,r)\subseteq
D:\,r\in\mathbb{R}_{+},x\in D\Big{\\}}$ and

$\displaystyle{\mathcal{X}}=\Cup_{x\in
D}\Big{\\{}B(x,r_{x}):B(x,r_{x})\subseteq D\Big{\\}}.$

By ${\mathcal{M}}({\mathcal{X}}\times{\mathcal{B}})$ we denote the space of
positive measures on the space ${\mathcal{X}}\times{\mathcal{B}}$ equipped
with $\tau-$ topology. Henceforth, we shall refer to ${\mathcal{X}}$ as
locally finite subset of the set $D$ and let ${\mathcal{N}}({\mathcal{B}})$ be
the set of counting numbers on ${\mathcal{B}}$ equipped with the discrete
topology. For any Boolean network $X^{o}$ we define a probability measure, the
_empirical mark measure_ ,
$L_{1}^{o}\in{\mathcal{M}}({\mathcal{X}}\times{\mathcal{B}})$, by

$L_{1}^{o}([x,b]):=\frac{1}{\lambda}\sum_{i\in I}\delta_{X_{i}^{o}}(x,b)$

and a symmetric finite measure, the _empirical connectivity measure_
$L_{2}^{o}\in{\mathcal{M}}({\mathcal{X}}\times{\mathcal{B}}\times{\mathcal{X}}\times{\mathcal{B}}),$
by

$L_{2}^{o}(x,b_{1},y,b_{2}):=\frac{1}{\lambda}\sum_{(i,j)\in
E}\Big{[}\delta_{(X_{i}^{o},X_{j}^{o})}+\delta_{(X_{j}^{o},X_{i}^{o})}\Big{]}(x,b_{1},y,b_{2}).$

Note that, while the empirical marked measure and the empirical coverage
measure are probability measure, total mass of the empirical pair measure is
$2|E|/\lambda^{2}$. Furthermore, $\|L_{1}^{o}\|$ is the number of devices
connected to device $i$.

In our model, we assume that each ball is small enough to contain a finite
number of users and that the position of one user does not coincide with the
other.

######  Proposition 2.1 (Probability of connection).

Let $\mathbb{X}=\left(X_{i}\right)_{i\in I},$ a Poisson Point Process with
intensity measure $\mu_{\lambda},$ which represents the configuration of
devices or users in a communication space $D\subseteq\mathbb{R}^{d}$ and
$Q_{\lambda}$ be the probability distribution of the volume of the balls
centered at $X_{i}.$ The probability $p_{\lambda}(x,b_{x},y,b_{y})$ that two
users located at $X_{i}=x$ and $X_{j}=y$, $i\neq j$, with coverage areas
$B(x,R_{x})$ and $B(y,R_{y})$ respectively are connected is given as

$\displaystyle
p_{\lambda}(x,b_{x},y,b_{y}):=\left[1-e^{-\mu_{\lambda}(b_{x}-b_{y})Q_{\lambda}\left(b_{x}\right)Q_{\lambda}\left(b_{y}\right)}\right],\;\;(x,b_{x},y,b_{2})\in{\mathcal{X}}\times\mathcal{B}\times{\mathcal{X}}\times\mathcal{B},$

where, $b_{x}=B(x,R_{x}),$ $b_{y}=B(y,R_{y}),$ $R_{x},R_{y}\in(0,\infty)$ and
$b_{x}-b_{y}=\left\\{z_{1}-z_{2}:z_{1}\in b_{1},z_{2}\in b_{2}\right\\}.$

###### Proof.

Let $X_{i}=x$ and $X_{j}=y$ be the location of two users in the communication
area $D.$ Also, let $B_{i}:=B(X_{i},R_{i})=b_{x}$ and
$B_{j}:=B(X_{j},R_{j})=b_{y}$ be the coverage areas of these two users. Let
$Q_{\lambda}$ be the probability distribution of the volume of the coverage
areas. The two users are connected when their coverage areas intersect. Now,
let $p_{\lambda}(x,b_{x},y,b_{y})$ be the probability that two users are
connected, given their locations. We have,

$\displaystyle p_{\lambda}(X_{i},B_{i},X_{j},B_{j})$
$\displaystyle=\mathbb{P}(B(X_{i},R_{i})\cap B(X_{j},R_{j})\neq\emptyset)$
(2.1) $\displaystyle=1-\mathbb{P}(B(X_{i},R_{i})\cap
B(X_{j},R_{j})=\emptyset)$
$\displaystyle=1-e^{-\mathbb{E}\left[\mu_{\lambda}(B_{i}-B_{j})\right]}$

Now, we have:

$\displaystyle\mathbb{E}[\mu_{\lambda}(B_{i}-B_{i})]$
$\displaystyle=\mathbb{E}[\mathbb{E}[\mu_{\lambda}(B_{i}-B_{j})|B_{j}]]$ (2.2)
$\displaystyle=\mathbb{E}\left[\int{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{B_{i}(b_{2})}\mu_{\lambda}(b_{2}-B_{j})Q_{\lambda}(db_{2})\right]$
$\displaystyle=\int{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{B_{j}(b_{1})}\int{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{B_{i}(b_{2})}\mu_{\lambda}(b_{2}-b_{1})Q_{\lambda}(db_{2})Q_{\lambda}(db_{1})$
$\displaystyle=\int\int{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{B_{j}(b_{1})}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{B_{i}(b_{2})}\mu_{\lambda}(b_{2}-b_{1})Q_{\lambda}(db_{2})Q_{\lambda}(db_{1})$
$\displaystyle=\mu_{\lambda}(B_{i}-B_{j})Q_{\lambda}(B_{i})Q_{\lambda}(B_{j})$

Therefore, the probability that two users are connected is,

$\displaystyle p_{\lambda}(X_{i},B_{i},X_{j},B_{j})$
$\displaystyle=1-e^{-\mu_{\lambda}(B_{i}-B_{j})Q_{\lambda}(B_{i})Q_{\lambda}(B_{j})}.$

∎

This result underlines the symmetric nature of this probability measure. In
fact, the probability
$p_{\lambda}(x,b_{x},y,b_{y})=p_{\lambda}(y,b_{y},x,b_{x})$ and this reflects
the fact that the event ”$X_{i}$ is connected to $X_{j}$ is the same as the
event ”$X_{j}$ is connected to $X_{i}$.” Also, it is clear that as the volume
of at least one of the two balls increases, the probability that the two balls
are connected increases and when their volume decreases, so does the
probability of connection. Now, we shall henceforth assume that as
$\lambda\rightarrow\infty,$

$\lambda^{-3}\mu_{\lambda}\rightarrow\mu\;\;\;\text{and}\;\;\;\lambda^{2}Q_{\lambda}\rightarrow
Q.$

Clearly, with these assumptions, we note that
$\mu_{\lambda}Q_{\lambda}Q_{\lambda}\approx Q\mu Q/\lambda,$ and the
probability of connection is again going to zero if the number of users is
increasing in the order of $\lambda^{3}$ but the strength of the signals
emanating from users, which is reflected in the volume of the balls is
decreasing at a rate $\lambda^{2}.$

We then perform a Taylor expansion around $0$ to obtain the expression;

$\displaystyle p_{\lambda}(X_{i},B_{i},X_{j},B_{j})$
$\displaystyle=1-\left[1-\mu_{\lambda}(B_{i}-B_{j})Q_{\lambda}(B_{i})Q_{\lambda}(B_{j})+O\left(\dfrac{1}{\lambda^{2}}\right)\right]$
$\displaystyle=\mu_{\lambda}(B_{i}-B_{j})Q_{\lambda}(B_{i})Q_{\lambda}(B_{j})+O\left(\dfrac{1}{\lambda^{2}}\right).$

Now, multiplying through by $\lambda,$ we get

$\displaystyle\lambda p_{\lambda}(X_{i},B_{i},X_{j},B_{j})$
$\displaystyle=\lambda\mu_{\lambda}(B_{i}-B_{j})Q_{\lambda}(X_{i})Q_{\lambda}(B_{j})+O\left(\dfrac{1}{\lambda}\right).$

Hence, taking the limit as $\lambda$ approaches $\infty$ we get

$\displaystyle\lambda
p_{\lambda}(X_{i},B_{i},X_{j},B_{j})\rightarrow\mu(B_{i}-B_{j})Q(B_{i})Q(B_{j}):=\Psi(B_{i},B_{j}),$

which proves the proposition. This implies that the asymptotic probability
density modelling two users that are connected is given by the product of the
probability distribution of the respective balls representing the coverage
areas of the users and the distribution of the number of users in the
intersection of the two coverage areas.

######  Proposition 2.2.

Let $A_{1},A_{2},\cdots,A_{n}$ be a decomposition of
$D\times\mathcal{B}\subset\mathbb{R}^{d}\times\mathcal{B}.$ Let $n<\lambda,$
since a Marked Poisson Point Process is locally finite, then the probability
distribution of the measure $L_{1}^{0}$ is bounded in the following manner:

$\displaystyle\sum_{i=1}^{n}\log\left[\dfrac{e^{-\mu_{\lambda}\otimes
Q_{\lambda}(A_{i})-\epsilon}\left(\mu_{\lambda}\otimes
Q_{\lambda}(A_{i})-\epsilon\right)^{\lambda\eta(A_{i})}}{\left(\lambda\eta(A_{i})\right)!}\right]\leq\log
P(L^{0}_{1}=\eta)$
$\displaystyle\leq\sum_{i=1}^{n}\log\left[\dfrac{e^{-\mu_{\lambda}\otimes
Q_{\lambda}(A_{i})+\epsilon}\left(\mu_{\lambda}\otimes
Q_{\lambda}(A_{i})+\epsilon\right)^{\lambda\eta(A_{i})}}{\left(\lambda\eta(A_{i})\right)!}\right]+\eta_{n},$
(2.3)

where,

$\lim_{n\rightarrow\infty}\lim_{\lambda\rightarrow\infty}\dfrac{1}{\lambda}\eta_{n}\left(\lambda,A_{1},\cdots,A_{n})\right)=0$

and the product measure is defined as

$\mu_{\lambda}\otimes
Q_{\lambda}(x,b_{x})=\mu_{\lambda}(x)Q_{\lambda}(b_{X}).$

###  2.2.  Main Results

######  Theorem 2.3.

Let $\mathbb{X}=\left(X_{i}\right)_{i\in I},$ be a Poisson Point Process with
intensity measure $\mu_{\lambda},$ which represents the configuration of
devices or users in a communication space $D\subseteq\mathbb{R}^{d}$ and
$Q_{\lambda}$ be the probability distribution of the volume of the balls
centered on the points $\mathbb{X}.$ i.e. coverage probability distribution.
Suppose that the intensity measure $\mu_{\lambda}:D\rightarrow[0,\infty)$
satisfies $\lambda^{-3}\mu_{\lambda}\rightarrow\mu$ and the coverage
probability distribution, $Q_{\lambda}:\mathcal{B}\rightarrow(0,1)$,
satisfying $\lambda^{2}Q_{\lambda}\rightarrow Q$. Then, as $\lambda\to\infty$,
the pair $(L_{1}^{0},L_{2}^{0})$ satisfies a large deviation principle in
$\mathcal{M}({\mathcal{X}}\times{\mathcal{B}})\times\mathcal{M}({\mathcal{X}}\times{\mathcal{B}}\times{\mathcal{X}}\times{\mathcal{B}})$
with speed $\lambda$ and rate function

$\displaystyle I(\omega,\pi)$ $\displaystyle=\begin{cases}H(\omega||\mu\otimes
Q)+\dfrac{1}{2}\Big{[}H(\pi||\Psi\omega\otimes\omega)+||\Psi\omega\otimes\omega||-||\pi||\Big{]}\;\;\,\,\text{
if\,\, $||\pi||<\infty,$}\\\ \infty,\;\;\text{otherwise}.\end{cases}$ (2.4)

######  Corollary 2.4.

Let $\mathbb{X}=\left(X_{i}\right)_{i\in I},$ be a Poisson Point Process with
intensity measure $\mu_{\lambda},$ which represents the configuration of
devices or users in a communication space $D\subseteq\mathbb{R}^{d}$ and
$Q_{\lambda}$ be the probability distribution of the volume of the balls
centered on the points $\mathbb{X}.$ i.e. coverage probability distribution.
Suppose that the intensity measure is given by
$\mu_{\lambda}(dx)=\lambda^{3}dx$ and the coverage probability distribution is
given by $Q_{\lambda}(db_{x})=\mbox{$\frac{4}{3}$}\pi
r^{3}/\lambda^{2}Vol(D)$. Then, as $\lambda\to\infty$, the pair
$(L_{1}^{0},L_{2}^{0})$ satisfies a large deviation principle in
$\mathcal{M}({\mathcal{X}}\times{\mathcal{B}})\times\mathcal{M}({\mathcal{X}}\times{\mathcal{B}}\times{\mathcal{X}}\times{\mathcal{B}})$
with speed $\lambda$ and rate function

$\displaystyle I(\omega,\varpi)$
$\displaystyle=\begin{cases}H(\omega||\mu\otimes
Q)+\dfrac{1}{2}\Big{[}H(\varpi||\Psi\omega\otimes\omega)+||\Psi\omega\otimes\omega||-||\varpi||\Big{]}\;\;\,\,\text{
if\,\, $||\varpi||<\infty,$}\\\ \infty,\;\;\text{otherwise},\end{cases}$ (2.5)

where

$\Psi(b_{x},b_{y})=\mbox{$\frac{16}{9}$}\pi^{2}r_{x}^{3}r_{y}^{3}\Big{[}\frac{Vol(b_{x}-b_{y})}{Vol(D)^{2}}\Big{]}.$

Hence, as $\lambda\to\infty$ the average number of connectivity per device
behaves as

$\|E\|/\lambda\to\mbox{$\frac{8\pi^{2}}{9Vol(D)^{2}}$}\int_{D\times{\mathcal{B}}}\int_{D\times{\mathcal{B}}}r_{x}^{3}r_{y}^{3}Vol(b_{x}-b_{y})Q(dbx)Q(dby)dxdy\,\,\,\mbox{
\bf{in probability}.}$

We first establish the Large Deviation Principle (LDP) for $L^{0}_{1}$ in
Section (3) with the appropriate rate function through the method of types.
Using the Gartner-Ellis theorem in Section (4), we prove that the measure
$L_{2}^{0}$ conditional on $L_{1}^{0}$ obeys an L.D.P, and we obtain the
solution of the rate function as a function of Entropy. Finally, we will prove
the LDP for the joint distribution of $L^{0}_{1}$ and $L^{0}_{2}$ in Section
(5) by employing the method of mixtures.

##  3\. Large deviation principle for the empirical marked measure
$L_{1}^{0}.$

In this section, we use the method of types to establish the LDP for the
empirical marked measure $L^{0}_{1}=\omega_{\lambda}$ as
$\lambda\rightarrow\infty.$ We begin by stating the large deviation
probability of $L^{0}_{1}=\omega_{\lambda}$ below:

######  Lemma 3.1.

Suppose $X^{o}$ is a Boolean random network with intensity measure
$\mu_{\lambda}:D\rightarrow[0,\infty)$ that satisfies
$\lambda^{-3}\mu_{\lambda}\rightarrow\mu$ and a coverage probability density
$Q_{\lambda}:\mathcal{B}\rightarrow(0,1)$, satisfying
$\lambda^{2}Q_{\lambda}\rightarrow Q$. Then, we have

$e^{-\lambda H\left(\eta^{(n)}||\mu^{(n)}\otimes
Q^{(n)}\right)+\gamma_{1}(\lambda)}\leq P(L^{0}_{1}=\eta)\leq e^{-\lambda
H\left(\eta^{(n)}||\mu^{(n)}\otimes Q^{(n)}\right)+\gamma_{2}(\lambda)}$ (3.1)

$\lim_{\lambda\rightarrow\infty}\gamma_{1}(\lambda)=0,\;\;\;\lim_{\lambda\rightarrow\infty}\gamma_{2}(\lambda)=\lim_{\lambda\rightarrow\infty}\frac{1}{\lambda}\eta_{n}(\lambda,A_{1},\dots,A_{n}),$
where $\eta^{(n)}$ and $\mu^{(n)}\otimes Q^{(n)}$ are the coarsening
projections of $\eta$ and $\mu\otimes Q$ on the decomposition
$(A_{1},\dots,A_{n}).$

To start the proof for Theorem 3.1, we will make use of Stirling’s formula:

###### Proof.

Assume $\lambda$ is large. We have from the upper bound in Equation (2.3)

$\displaystyle\log P(L_{1}^{0}=\eta)$
$\displaystyle\leq\sum_{j=1}^{n}\left\\{{-\lambda\mu\otimes
Q(A_{j})}+\lambda\eta(A_{j})\log\left[\lambda\mu\otimes
Q(A_{j})\right]-\log\left[\lambda\eta(A_{j})\right]!\right\\}+\eta_{n}(\lambda,A_{1},\dots,A_{n})$

Using the upper bound of equation (3.1), we get,

$\displaystyle\log P(L_{1}^{0}=\eta)$
$\displaystyle\leq\sum_{j=1}^{n}\left\\{-\lambda\mu\otimes
Q(A_{j})-\log\left[(2\pi)^{\frac{1}{2}}(\lambda\eta(A_{j}))^{\lambda\eta(A_{j})+\frac{1}{2}}e^{-\lambda\eta(A_{j})}\right]\right\\}$
$\displaystyle\;\;\;\;\;+\sum_{j=1}^{n}\left\\{\dfrac{1}{12\lambda\eta(A_{j})+1}+\lambda\eta(A_{j})\log\left[\lambda\mu\otimes
Q(A_{j})\right]\right\\}+\eta_{n}(\lambda,A_{1}.\dots,A_{n})$
$\displaystyle\log P(L_{1}^{0}=\eta)$
$\displaystyle\leq\sum_{j=1}^{n}\left\\{-\lambda\mu\otimes
Q(A_{j})-\dfrac{1}{2}\log(2\pi)-\left[\lambda\eta(A_{j})+\frac{1}{2}\right]\log\left[\lambda\eta(A_{j})\right]+\lambda\eta(A_{j})\right\\}$
$\displaystyle\;\;\;\;\;+\sum_{j=1}^{n}\left\\{\dfrac{1}{12\lambda\eta(A_{j})+1}+\lambda\eta(A_{j})\log\left[\lambda\mu\otimes
Q(A_{j})\right]\right\\}+\eta_{n}(\lambda,A_{1},\dots,A_{n})$
$\displaystyle\log P(L_{1}^{0}=\eta)$
$\displaystyle\leq\sum_{i=1}^{n}\left\\{-\lambda\left[\mu\otimes
Q(A_{j})-\eta(A_{j})\right]-\lambda\eta(A_{j})\log\dfrac{\eta(A_{j})}{\mu\otimes
Q(A_{j})}-\dfrac{1}{2}\log[\lambda\eta(A_{j})]\right\\}$
$\displaystyle\;\;\;\;\;\;+\sum_{j=1}^{n}\left\\{\dfrac{1}{12\lambda\eta(A_{j})+1}-\dfrac{1}{2}\log(2\pi)\right\\}+\eta_{n}(\lambda,A_{1},\dots,A_{n})$
$\displaystyle\log P(L_{1}^{0}=\eta)$
$\displaystyle\leq\sum_{i=1}^{n}\left\\{-\lambda\left[\mu\otimes
Q(A_{j})-\eta(A_{j})\right]-\lambda\eta(A_{j})\log\dfrac{\lambda\eta(A_{j})}{\mu\otimes
Q(A_{j})}\right\\}$
$\displaystyle\;\;\;\;\;\;-\sum_{j=1}^{n}\lambda\left\\{\dfrac{\log[\eta(A_{j})]}{2\lambda}-\dfrac{1}{12\lambda^{2}\eta(A_{j})+\lambda}+\dfrac{\log(2\pi)}{2\lambda}+\dfrac{\eta_{n}(\lambda,A_{1},\dots,A_{n})}{\lambda}\right\\}$

We then choose $\gamma 2(\lambda)$ as

$\displaystyle\gamma_{2}(\lambda)$
$\displaystyle=\dfrac{\log[\lambda\eta(A_{j})]}{2\lambda}-\dfrac{1}{12\lambda^{2}\eta(A_{j})+\lambda}+\dfrac{\log(2\pi)}{2\lambda}+\dfrac{\eta_{n}(\lambda,A_{1},\dots,A_{n})}{\lambda}.$

We have,

$\displaystyle\lim_{\lambda\rightarrow\infty}\gamma_{2}(\lambda)$
$\displaystyle=\lim_{\lambda\rightarrow\infty}\left[\dfrac{\log[\lambda\eta(A_{j})]}{2\lambda}-\dfrac{1}{12\lambda^{2}\eta(A_{j})+\lambda}+\dfrac{\log(2\pi)}{2\lambda}+\dfrac{\eta_{n}(\lambda,A_{1},\dots,A_{n})}{\lambda}\right]$
$\displaystyle=\lim_{\lambda\rightarrow\infty}\dfrac{1}{\lambda}\eta_{n}(\lambda,A_{1},\dots,A_{n}),$

and this concludes the proof for the upper bound of Theorem (3.1). Again,
suppose $\lambda$ is large, we obtain from the lower bound of equation (2.3),

$\displaystyle\log P(L_{1}^{0}=\eta)$
$\displaystyle\geq\sum_{j=1}^{n}\left\\{{-\lambda\mu\otimes
Q(A_{j})}+\lambda\eta(A_{j})\log\left[\lambda\mu\otimes
Q(A_{j})\right]-\log\left[\lambda\eta(A_{j})\right]!\right\\}.$

Using the lower bound of equation (3.1), we get,

$\displaystyle\log P(L_{1}^{0}=\eta)$
$\displaystyle\geq\sum_{j=1}^{n}\left\\{-\lambda\mu\otimes
Q(A_{j})-\log\left[(2\pi)^{\frac{1}{2}}(\lambda\eta(A_{j}))^{\lambda\eta(A_{j})+\frac{1}{2}}e^{-\lambda\eta(A_{j})}\right]\right\\}$
$\displaystyle\;\;\;\;\;+\sum_{j=1}^{n}\left\\{\dfrac{1}{12\lambda\eta(A_{j})}+\lambda\eta(A_{j})\log\left[\lambda\mu\otimes
Q(A_{j})\right]\right\\}$ $\displaystyle\log P(L_{1}^{0}=\eta)$
$\displaystyle\geq\sum_{j=1}^{n}\left\\{-\lambda\mu\otimes
Q(A_{j})-\dfrac{1}{2}\log(2\pi)-\left[\lambda\eta(A_{j})+\frac{1}{2}\right]\log\left[\lambda\eta(A_{j})\right]+\lambda\eta(A_{j})\right\\}$
$\displaystyle\;\;\;\;\;+\sum_{j=1}^{n}\left\\{\dfrac{1}{12\lambda\eta(A_{j})}+\lambda\eta(A_{j})\log\left[\lambda\mu\otimes
Q(A_{j})\right]\right\\}$ $\displaystyle\log P(L_{1}^{0}=\eta)$
$\displaystyle\geq\sum_{i=1}^{n}\left\\{-\lambda\left[\mu\otimes
Q(A_{j})-\eta(A_{j})\right]-\lambda\eta(A_{j})\log\dfrac{\eta(A_{j})}{\mu\otimes
Q(A_{j})}-\dfrac{1}{2}\log[\lambda\eta(A_{j})]\right\\}$
$\displaystyle\;\;\;\;\;\;+\sum_{j=1}^{n}\left\\{\dfrac{1}{12\lambda\eta(A_{j})}-\dfrac{1}{2}\log(2\pi)\right\\}$
$\displaystyle\log P(L_{1}^{0}=\eta)$
$\displaystyle\geq\sum_{i=1}^{n}\left\\{-\lambda\left[\mu\otimes
Q(A_{j})-\eta(A_{j})\right]-\lambda\eta(A_{j})\log\dfrac{\eta(A_{j})}{\mu\otimes
Q(A_{j})}\right\\}$
$\displaystyle\;\;\;\;\;\;-\sum_{j=1}^{n}\lambda\left\\{\dfrac{\log[\lambda\eta(A_{j})]}{2\lambda}-\dfrac{1}{12\lambda^{2}\eta(A_{j})}+\dfrac{\log(2\pi)}{2\lambda}\right\\}.$

We then choose $\gamma_{1}(\lambda)$ as

$\displaystyle\gamma_{1}(\lambda)$
$\displaystyle=\dfrac{\log[\lambda\eta(A_{j})]}{2\lambda}-\dfrac{1}{12\lambda^{2}\eta(A_{j})}+\dfrac{\log(2\pi)}{2\lambda}$

We have,

$\displaystyle\lim_{\lambda\rightarrow\infty}\gamma_{1}(\lambda)$
$\displaystyle=\lim_{\lambda\rightarrow\infty}\left[\dfrac{\log[\lambda\eta(A_{j})]}{2\lambda}-\dfrac{1}{12\lambda^{2}\eta(A_{j})}+\dfrac{\log(2\pi)}{2\lambda}\right]=0$

and this concludes the proof for the lower bound of Theorem (3.1). ∎

######  Lemma 3.2.

Suppose $X^{o}$ is a Boolean random network with intensity measure
$\mu_{\lambda}:D\rightarrow[0,\infty)$ that satisfies
$\lambda^{-3}\mu_{\lambda}\rightarrow\mu$ and a coverage probability density
$Q_{\lambda}:\mathcal{B}\rightarrow(0,1)$ , satisfying
$\lambda^{2}Q_{\lambda}\rightarrow Q$. Then, for large $\lambda,$ we have

$|I|\leq 2\lambda\;\;\;\text{almost surely}.$

###### Proof.

Let $A_{1},A_{2},\cdots,A_{m}$ be a disjoint decomposition of $\mathbb{R}^{d}$
such that $|{\mathcal{X}}\cap A_{k}|<a_{k}<\infty,$ for
$a_{k}(A_{k}):=a_{k}\in\mathbb{R}.$ Let
$a=\max(a_{1},a_{2},a_{2},a_{3},...,a_{m})$ and observe that,
$|I|=\sum_{k=1}^{m}|I_{k}|,$ where $I_{k}={\mathcal{X}}\cap A_{k}$.Note
$|I_{1}|,|I_{2}|,|I_{3}|,...,|I_{m}|$ are independent Poisson distributed
random variables with mean and variance $\mu({\mathcal{X}}\cap A_{k}),$
$k=1,2,3,...,m$, respectively. Observe that $I_{k}\leq a,$ for all
$k=1,2,\dots,m$ and hence, by applying the Bennett’s inequality to the
sequence, $I_{1},I_{2},\dots,I_{m},$ we have

$\displaystyle\mathbb{P}\left(|I|-\mathbb{E}|I|>\lambda\right)\leq\exp\\{-\frac{\lambda}{a^{2}}\Phi(a)\\},$
(3.2)

where $Vol(D)$ means the volume of the geometrical space $D$ and
$\Phi(u)=(1+u)\log(1+u)-u.$ Now, using Equation (3.2), we have

$P\left(|I|\leq\mathbb{E}|I|+\lambda\right)\geq
1-\exp\\{-\frac{\lambda}{a^{2}}\Phi(a)\\},$

and this leads to

$\lim_{\lambda\rightarrow\infty}\mathbb{P}\left(|I|\leq 2\lambda\right)\geq
1.$

Therefore, $|I|\leq 2\lambda$ almost surely, which ends the proof of Lemma
(3.2). ∎

######  Lemma 3.3 (LDP for $L_{1}^{0}$).

Suppose $X^{o}$ is a Boolean random network with intensity measure
$\mu_{\lambda}:D\rightarrow[0,\infty)$ that satisfies
$\lambda^{-3}\mu_{\lambda}\rightarrow\mu$ and a coverage probability density
$Q_{\lambda}:\mathcal{B}\rightarrow(0,1)$ , satisfying
$\lambda^{2}Q_{\lambda}\rightarrow Q$. Then, as $\lambda\rightarrow\infty,$
$L_{1}^{0}$ obeys an LDP with rate function:

$\displaystyle I_{1}(\omega)$
$\displaystyle=\begin{cases}{H}(\omega||\mu\otimes
Q)\;\;\text{if}\;\;||\omega||=1,\\\ \\\
\infty,\;\;\;\text{otherwise.}\end{cases}$ (3.3)

The proof of the above theorem follows the arguments of [17].

###### Proof.

Let
$\mathcal{M}_{\lambda}\left(\mathcal{X}\times\mathcal{B}\right):=\left\\{\eta\in\mathcal{M}\left(\mathcal{X}\times\mathcal{B}\right):\lambda\eta(a)\in\mathbb{N}\;\;\text{for
all}\;\;a\in\mathcal{X}\right\\}$ and let $S$ be a subset of
$\mathcal{M}(\mathcal{X\times B}).$ We write
$\beta_{n}:=\max\left(|\mathcal{X}\times B_{1}|,|\mathcal{X}\times
B_{2}|,\dots,|\mathcal{X}\times B_{n}|\right)$ and note that
$|\mathcal{X}\times B|<\infty$, for all $i=1,2,\dots,n$ by construction. Using
Theorem (3.1) and Lemma (3.2), we obtain:

$\displaystyle(1+\lambda+\lambda)^{-n\beta_{n}}e^{-\lambda\inf_{\\{\eta\in
F^{0}\cap\mathcal{M}(\mathcal{X\times B})\\}}H(\eta^{(n)}||\mu^{(n)}\otimes
Q^{(n)}+\theta_{1}(\lambda)}$ $\displaystyle\leq\sum_{\eta\in
F^{0}\cap\mathcal{M}(\mathcal{X\times B})}e^{-\lambda H(\eta^{(n)}||\mu\otimes
Q^{(n)})+\theta_{2}(\lambda)}$ $\displaystyle\leq\mathbb{P}(L_{1}^{0}\in F)$
$\displaystyle\leq\sum_{\eta\in cl(F)\cap\mathcal{M}(\mathcal{X})}e^{-\lambda
H(\eta^{(n)}||\mu\otimes Q^{(n)})+\theta_{2}(\lambda)}$
$\displaystyle\leq(1+\lambda+\lambda)^{n\beta_{n}}e^{-\lambda
K+\theta_{2}(\lambda)},$

where

$K=\inf_{\eta\in cl(F)\cap\mathcal{M}(\mathcal{X\times
B})}{H}(\eta^{(n)}||\mu^{(n)}\otimes Q^{(n)}),$

$\eta^{(n)}$ and $\mu^{(n)}\otimes Q^{(n)}$ are the coarsening projections of
$\eta$ and $\mu\otimes Q$ on the decomposition $(A_{1},\dots,A_{n}).$

Taking the limit as $\lambda\rightarrow\infty,$ we have

$\displaystyle\liminf_{\lambda\rightarrow\infty}\left\\{-\inf_{\\{\eta\in
F^{0}\cap\mathcal{M}_{\lambda}(\mathcal{X})\\}}\,H(\eta^{(n)}||\mu\otimes
Q^{(n)})\right\\}$
$\displaystyle\leq\lim_{\lambda\rightarrow\infty}\dfrac{1}{\lambda}\log\mathbb{P}\left(L_{1}^{0}\in
F\right)$
$\displaystyle\leq\limsup_{\lambda\rightarrow\infty}\left\\{-\inf_{\\{\eta\in
cl(F)\cap\mathcal{M}_{\lambda}(\mathcal{X})\\}}\,H(\eta^{(n)}||\mu\otimes
Q^{(n)})\right\\}.$

We then observe $cl(F)\cap\mathcal{M}_{\lambda}(\mathcal{X\times X})\subset
cl(F)$ for all $\lambda\in\mathbb{R}_{+}$ and hence we have

$\displaystyle\limsup_{\lambda\rightarrow\infty}\left\\{-\inf_{\\{\eta\in
cl(F)\cap\mathcal{M}_{\lambda}(\mathcal{X})\\}}\,H(\eta^{(n)}||\mu\otimes
Q^{(n)})\right\\}$ $\displaystyle\leq-\inf_{\\{\eta\in
cl(F)\\}}\,H\left(\eta^{(n)}||\mu^{(n)}\otimes Q^{(n)}\right).$

Using the arguments of [8, page 17], we obtain

$\displaystyle\liminf_{\lambda\rightarrow\infty}\left\\{-\inf_{\\{\eta\in
F^{0}\cap\mathcal{M}_{\lambda}(\mathcal{X\times
B})\\}}\,H(\eta^{(n)}||\mu\otimes Q^{(n)})\right\\}$
$\displaystyle\geq-\inf_{{\eta\in F^{0}}}\,H\left(\eta^{(n)}||\mu^{(n)}\otimes
Q^{(n)}\right).$

Therefore, we have

$\displaystyle-\inf_{{\eta\in F^{0}}}\,H\left(\eta^{(n)}||\mu^{(n)}\otimes
Q^{(n)}\right)$
$\displaystyle\leq\lim_{\lambda\rightarrow\infty}\dfrac{1}{\lambda}\log\mathbb{P}\left(L_{1}^{0}\in
F\right)\leq-\inf_{{\eta\in cl(F)}}\,H\left(\eta^{(n)}||\mu^{(n)}\otimes
Q^{(n)}\right),$

where $\eta^{(n)}$ and $\mu^{(n)}\otimes Q^{(n)}$ are the coarsening
projections of $\eta$ and $\mu\otimes Q$ on the decomposition
$(A_{1},\dots,A_{n}).$ Now, when $n\rightarrow\infty,$ we have

$\displaystyle-\inf_{{\eta\in F^{0}}}\,H\left(\eta||\mu\otimes Q\right)$
$\displaystyle\leq\lim_{\lambda\rightarrow\infty}\dfrac{1}{\lambda}\log\mathbb{P}\left(L_{1}^{0}\in
F\right)\leq-\inf_{{\eta\in cl(F)}}\,H\left(\eta||\mu\otimes Q\right),$

which concludes the proof of Theorem (3.3). ∎

We have established the LDP of the empirical marked measure, $L^{0}_{1}.$ We
then establish in the following section the LDP of the empirical connectivity
measure $L^{0}_{2}$ given $L^{0}_{1}=\omega.$

##  4\. LDP for the empirical connectivity measure $L^{0}_{2}$ given
$L_{1}^{0}.$

######  Lemma 4.1.

Let $\mathbb{X}=\left(X_{i}\right)_{i\in I},$ be a Poisson Point Process with
intensity measure $\mu_{\lambda},$ which represents the configuration of
devices or users in a communication space $D\subseteq\mathbb{R}^{d}$ and
$Q_{\lambda}$ be the probability distribution of the volume of the balls
centered on the points $\mathbb{X}.$ i.e. coverage probability distribution.
Suppose the probability of connection $p_{\lambda}(b_{X},b_{y})$ of two users
having their location in $\mathbb{X}$ satisfies $\lambda
p_{\lambda}(b_{X},b_{Y})\rightarrow\mu(b_{X}-b_{Y})Q(b_{X})Q(b_{Y}).$ Then
$L^{0}_{2}$ conditional on the event $\\{$ $L^{0}_{1}=\omega_{\lambda}\\}$
satifies a large deviation principle with rate function given as,

$\displaystyle I_{\omega}(\pi)$
$\displaystyle=\begin{cases}\dfrac{1}{2}\Bigg{[}H(\pi||\Psi\omega\otimes\omega)+||\Psi\omega\otimes\omega||-||\pi||\Bigg{]},&\text{if}\;\;||\pi||<\infty\\\
\\\ \infty,&\text{otherwise.}\end{cases}$ (4.1)

The main method used in this section is the Gartner-Ellis Theorem. We
establish the LDP for the empirical connectivity measure $L^{0}_{2}$ given the
empirical marked measure $L^{0}_{1}=\omega_{\lambda}$ as
$\lambda\rightarrow\infty.$ From the definition of these two measures in
Section 2, it is evident that $\lambda L^{0}_{2}(b_{X},b_{Y}$ conditional on
$L^{0}_{1}(b_{x})=\omega_{\lambda}(b_{X})$ follows a Binomial distribution
with parameters,
$\dfrac{\lambda^{2}}{2}\omega_{\lambda}(b_{X})\omega_{\lambda}(b_{Y})$ and
$P_{\lambda}(b_{X},b_{Y})$.

######  Lemma 4.2.

Let $\mathbb{X}=\left(X_{i}\right)_{i\in I},$ a P.P.P with intensity measure
$\mu_{\lambda},$ represent the configuration of devices or users in a
communication space $D\subseteq\mathbb{R}^{d}$ and $Q_{\lambda}$ be the
probability distribution of the volume of the balls centered at $X_{i}.$
Suppose the probability of connection $p_{\lambda}(b_{X},b_{y})$ of two users
having their location in $\mathbb{X}$ satisfies $\lambda
p_{\lambda}(b_{X},b_{Y})\rightarrow\mu(b_{X}-b_{Y})Q(b_{X})Q(b_{Y}).$
Furthermore, let $L^{0}_{2}$ be conditional on the event $\\{$
$L^{0}_{1}=\omega_{\lambda}\\}$. Now, let
$g:\mathcal{B}\times\mathcal{B}\rightarrow\mathbb{R}$ be a bounded function.
Then,

$\displaystyle\lim_{\lambda\rightarrow\infty}\dfrac{1}{\lambda}\log\Phi_{\lambda}(g)=\Phi(g),$
(4.2)

where,

$\Phi(g)=-\dfrac{1}{2}\int_{b_{X}\in(D\times\mathcal{B})}\int_{b_{Y}\in(D\times\mathcal{B})}(1-e^{g(b_{X},b_{Y})})\Psi(b_{X},b_{Y})\omega_{\lambda}(db_{X})\omega_{\lambda}(db_{Y}).$

###### Proof.

We assume $\Phi_{\lambda}(g)$ is the conditional m.g.f of $L_{2}^{0}.$ By
definition, we have:

$\displaystyle\Phi_{\lambda}(g)$
$\displaystyle=\mathbb{E}\left[e^{\lambda\langle
g,L^{2}_{0}\rangle}|L^{0}_{1}=\omega_{\lambda}\right]$
$\displaystyle=\mathbb{E}\left[e^{\lambda\int\int
g(b_{X},b_{Y})L^{2}_{0}(db_{X},db_{Y})}|L^{0}_{1}=\omega_{\lambda}\right].$

Now, let $A_{1},\dots,A_{n}$ be a decomposition of
$\left(D\times\mathcal{B}\right)^{2}$ into locally finite subsets. We then
write:

$\displaystyle\Phi_{\lambda}(g)$
$\displaystyle=\mathbb{E}\left[\prod_{i=1}^{n}\prod_{(b_{X},b_{Y})\in
A_{j}}e^{\lambda
g(b_{X},b_{Y})L^{2}_{0}(db_{X},db_{Y})}|L^{0}_{1}=\omega_{\lambda}\right].$

But the balls are independent of each other, hence we obtain:

$\displaystyle\Phi_{\lambda}(g)$
$\displaystyle=\prod_{i=1}^{n}\prod_{(b_{X},b_{Y})\in
A_{j}}\mathbb{E}\left[e^{\lambda
g(b_{X},b_{Y})L^{2}_{0}(db_{X},db_{Y})}|L^{0}_{1}=\omega_{\lambda}\right].$

At this point, we observe that the conditional distribution of $L_{2}^{0}$
given $L_{1}^{0}=\omega_{\lambda}$ is binomial, i.e,

$L^{0}_{2}|L^{0}_{1}=\omega_{\lambda}\sim
Bin\left(\dfrac{\lambda^{2}}{2}\omega_{\lambda}(b_{X})\omega_{\lambda}(b_{Y}),P_{\lambda}(b_{X},b_{Y})\right).$

Also, the conditional expectation in the previous expression is just the
conditional m.g.f of $L_{2}^{0}.$ We then write:

$\displaystyle\Phi_{\lambda}(g)$
$\displaystyle=\prod_{i=1}^{n}\prod_{(b_{X},b_{Y})\in
A_{j}}\left[1-p_{\lambda}(b_{X},b_{Y})+p_{\lambda}(b_{X},b_{Y})e^{g(b_{X},b_{Y})}\right]^{\frac{\lambda^{2}}{2}\omega_{\lambda}(db_{X})\omega_{\lambda}(db_{Y})}.$

Now,

$\displaystyle\log\Phi_{\lambda}(g)$
$\displaystyle=\sum_{i=1}^{n}\int_{(b_{X},b_{Y})\in
A_{j}}\log\left[1-p_{\lambda}(b_{X},b_{Y})+p_{\lambda}(b_{X},b_{Y})e^{g(b_{X},b_{Y})}\right]^{\frac{\lambda^{2}}{2}\omega_{\lambda}(db_{X})\omega_{\lambda}(db_{Y})}$
$\displaystyle=\sum_{i=1}^{n}{\int}_{(b_{X},b_{Y})\in
A_{j}}\frac{\lambda^{2}}{2}\omega_{\lambda}(db_{X})\omega_{\lambda}(db_{Y})\log\left[1-p_{\lambda}(b_{X},b_{Y})+p_{\lambda}(b_{X},b_{Y})e^{g(b_{X},b_{Y})}\right]$
$\displaystyle=\dfrac{1}{2}\sum_{i=1}^{n}\int_{(b_{X},b_{Y})\in
A_{j}}\log\left[1-p_{\lambda}\left(b_{X},b_{Y}\right)\left(1-e^{g(b_{X},b_{Y})}\right)\right]\lambda^{2}\omega_{\lambda}(db_{X})\omega_{\lambda}(db_{Y}).$

Using the Taylor expansion of
$\log(1-x)=-x-\dfrac{x^{2}}{2}-\dfrac{x^{3}}{3}-\cdots$, we get,

$\displaystyle=\dfrac{1}{2}\sum_{i=1}^{n}\int_{(b_{X},b_{Y})\in
A_{j}}\left[-p_{\lambda}(b_{X},b_{Y})(1-e^{g(b_{X},b_{Y})})+O\left(\dfrac{1}{\lambda^{2}}\right)\right]\lambda^{2}\omega_{\lambda}(db_{X})\omega_{\lambda}(db_{Y})$

Diving through by $\lambda,$ we get,

$\displaystyle\dfrac{1}{\lambda}\log\Phi_{\lambda}(g)$
$\displaystyle=-\dfrac{1}{2}\sum_{i=1}^{n}\int_{(b_{X},b_{Y})\in
A_{j}}\left[-\lambda
p_{\lambda}(b_{X},b_{Y})(1-e^{g(b_{X},b_{Y})})+O\left(\dfrac{1}{\lambda}\right)\right]\omega_{\lambda}(db_{X})\omega_{\lambda}(db_{Y})$

When $\lambda\rightarrow\infty,$ and with the assumption that
$\omega_{\lambda}\rightarrow w$, we obtain:

$\displaystyle\lim_{\lambda\rightarrow\infty}\dfrac{1}{\lambda}\log\Phi_{\lambda}(g)$
$\displaystyle=-\dfrac{1}{2}\sum_{i=1}^{\infty}\int_{(b_{X},b_{Y})\in
A_{j}}(1-e^{g(b_{X},b_{Y})})k(b_{X},b_{Y})\omega_{\lambda}(db_{X})\omega_{\lambda}(db_{Y})$
$\displaystyle=-\dfrac{1}{2}\int_{(D\times\mathcal{B})^{2}}(1-e^{g(b_{X},b_{Y})})k(b_{X},b_{Y})\omega_{\lambda}(db_{X})\omega_{\lambda}(db_{Y})$
$\displaystyle=-\dfrac{1}{2}\int_{b_{X}\in(D\times\mathcal{B})}\int_{b_{Y}\in(D\times\mathcal{B})}(1-e^{g(b_{X},b_{Y})})k(b_{X},b_{Y})\omega_{\lambda}(db_{X})\omega_{\lambda}(db_{Y})$
$\displaystyle=-\dfrac{1}{2}\int_{b_{X}\in(D\times\mathcal{B})}\int_{b_{Y}\in(D\times\mathcal{B})}(1-e^{g(b_{X},b_{Y})})k\omega_{\lambda}\otimes\omega_{\lambda}(db_{X},db_{Y})$

∎

which concludes the proof of Lemma (4.2).

It is clear that the function $\Phi$ is differentiable. Therefore, by the
Gartner-Ellis theorem, $L_{2}^{0}$ conditional on
$\left\\{L_{1}^{0}=\omega_{\lambda}\right\\}$ obeys a large deviation
principle with speed $\lambda$ and the variational formulation of the rate
function:

$\displaystyle I_{w}(\pi)$ $\displaystyle=\dfrac{1}{2}\sup_{g}\left\\{\langle
g,\pi\rangle-\Phi(g)\right\\}$ (4.3)

The solution to the rate function is given in terms of entropy as:

$\displaystyle I_{w}(\pi)$
$\displaystyle=\begin{cases}\dfrac{1}{2}\Bigg{[}\,H(\pi||kw\otimes
w)+||kw\otimes w||-||\pi||\Bigg{]}&\text{if}\;\;||\pi||<\infty\\\ \\\
\infty,&\text{otherwise}\end{cases}$ (4.4)

where,

$\,H(\pi||\mu)=\sum\pi\log\dfrac{\pi}{\mu}.$

We shall note that the rate function obtained in Equation (4.3) is the
Legendre transform and hence is a convex function.

##  5\. Joint Large deviation principle for the empirical pair and marked
measures.

In this section, the method of mixtures is employed. We follow the arguments
made in [17] and [19].

For $\lambda\in\mathbb{R}_{+},$ we define

$\mathcal{M}_{\lambda}(\mathcal{X}):=\left\\{w\in\mathcal{M}(\mathcal{X}):\lambda
w(a)\in\mathbb{N}\;\;\text{for all}\;\;a\in\mathcal{X}\right\\},$
$\mathcal{M}_{\lambda}(\mathcal{X\times
X}):=\left\\{\pi\in\mathcal{M}(\mathcal{X\times
X}):\lambda\pi(a,b)\in\mathbb{N}\;\;\text{for all}\;\;(a,b)\in\mathcal{X\times
X}\right\\}.$

Again we denote by $\Theta_{\lambda}:=\mathcal{M}_{\lambda}(\mathcal{X})$ and
$\Theta:=\mathcal{M}(\mathcal{X}).$

Finally, we make these two definitions,

$P^{(\lambda)}_{\omega_{\lambda}}\left(\eta_{\lambda}\right):=\mathbb{P}\left(L^{0}_{2}=\eta_{\lambda}|L_{1}^{0}=\omega_{\lambda}\right)$

and

$P^{(\lambda)}(\omega_{\lambda}):=\mathbb{P}\left(L_{1}^{0}=\omega_{\lambda}\right),$

then the joint distribution of $L^{0}_{1}$ and $L^{0}_{2}$ is the mixture of
$P^{(\lambda)}_{\omega_{\lambda}}$ with
$P^{(\lambda)}(\omega_{\lambda}):=\mathbb{P}\left(L_{1}^{0}=\omega_{\lambda}\right)$
defined as

$\displaystyle
d\tilde{P}^{\lambda}(\omega_{\lambda},\eta_{\lambda}):=dP^{(\lambda)}_{\omega_{\lambda}}\left(\eta_{\lambda}\right)dP^{(\lambda)}\left(\omega_{\lambda}\right).$

[3] provides the criteria for the existence of large deviation principles for
mixtures and the goodness of the rate function if individual large deviation
principles have been established.

The lemmas below make certain the validity of large deviation principles for
the mixtures and for the goodness of the rate function if individual large
deviation principles are known.

We note that the family of measures $(P^{\lambda}:\lambda\in(0,\infty))$ is
exponentially tight on $\Theta$.

######  Lemma 5.1.

The family of measures $(\tilde{P}:\lambda\in\mathbb{R}_{+})$ is exponentially
tight on $\Theta\times\mathcal{M}(\mathcal{X\times X}$.

###### Proof.

Let $|E|$ be the cardinality of the edge set $E$ and $t=1-(1-e^{l})e^{-\eta}.$
Then using Chebyshev’s inequality and Lemma (4.2), we get for sufficiently
large $\lambda$,

$\displaystyle\mathbb{P}\left(|E|\leq\lambda^{2}l\right)$ $\displaystyle\leq
e^{-\lambda^{2}l}\mathbb{E}\left[e^{|E|}\right]$ $\displaystyle\leq
e^{-\lambda^{2}l}\sum_{i=0}^{\infty}\sum_{k=0}^{i}e^{k}{i\choose
k}(e^{-\eta})^{k}(1-e^{-\eta})^{i-k}\dfrac{e^{-\lambda}\lambda^{i}}{i!}$
$\displaystyle\leq e^{-\lambda^{2}l}e^{-\lambda}e^{t\lambda}.$

Given $s\in\mathbb{N},$ we choose $s>q$ and observe that for sufficiently
large $\lambda$, we have

$\displaystyle\mathbb{P}\left(|E|\leq\lambda^{2}s\right)\leq
e^{-\lambda^{2}s},$

Therefore, we obtain

$\displaystyle\mathbb{P}\left(|E|\leq\lambda^{2}s/2\right)\leq
e^{-\lambda^{2}s/2},$

and Lemma (5.1) is then proved. ∎

Recall the definition of the relative entropy and observe that it is a lower
semi-continuous function.

######  Lemma 5.2.

$I_{w}$ are lower semi-continuous functions.

###### Proof.

Note that the relative entropy, $H(\pi|kw\otimes w),$ is a lower semi-
continuous function on the space $\mathcal{M}_{\lambda}(\mathcal{X\times X})$
and $||kw\otimes w||-||\pi||$ is a linear function in $\pi$. As $I_{w}$ is a
function of a relative entropy plus a linear function, one can conclude that
$I_{w}$ is lower semi-continuous.

∎

Using [3, Theorem 5(b)], the two previous lemmas; Lemma 5.1, Lemma 5.2 and the
two LDPs we have proved; Theorem 3.3 and Theorem 4.1 ensure that under
$(\tilde{P}^{\lambda})$ the random variables $(L_{1}^{0},L_{2}^{0})$ satisfy a
large deviation principle on
$\mathcal{M}_{\lambda}(\mathcal{X})\times\mathcal{M}_{\lambda}(\mathcal{X\times
X})$ with good rate function $I$ which ends the proof of Theorem 2.3.

##  6\. Conclusion

In this final section, we present the summary, the key findings and a
conclusion to the article. Some few recommendations are made which can be the
starting point for future research works.

The paramount aim of this article is to determine the joint large deviation
principle for the empirical pair measure and the empirical marked measure in a
telecommunication system using the Boolean model. To be able to achieve this
objective, it is of prime importance to achieve some required goals. We, first
and foremost, generated the position of users in a communication area via a
Poisson point process and provided the boolean model settings. We then, after
finding the asymptotic behavior of the probability that two users are
connected, proceeded to find the large deviation principle of the marginal law
of the empirical marked measure and the conditional law of the empirical pair
measure given the marginal law measure. Finally, the method of mixtures was
employed to formulate the joint large deviation principles of the empirical
measures.

The rate function is the most important part of the large deviation, as it
characterizes rare behaviors that may occur in the telecommunication system
and the most likely way an unlikely event will occur. It provides a bound on
the probability that such rare events may happen. An example of such events is
designated by the term frustration events which involve very bad service
quality. The rate function provides a way to understand and control such
events. Further, it highlights on how the major components of the
telecommunication system, namely: the users and the coverage area, relate in
the whole system. The rate function of the joint LDP of the empirical marked
and connectivity measures $(L^{0}_{1},L^{0}_{2})$ is non-negative.

This work states that the pair measure $(L^{0}_{1},L^{0}_{2})$ of the spatial
telecommunication system which is represented by a Boolean network obeys a
large deviation with speed $\lambda$ and large deviation probability

$\displaystyle P\left[(L^{0}_{1},L^{0}_{2})=(w,\pi)\right]\approx e^{-\lambda
I(w,\pi)},$ (6.1)

where the rate function $I(w,\pi)$ is in terms of relative entropies
$\,H(w||\mu\otimes Q),\,H(\pi||kw\otimes w)$ and a linear function $||\pi||.$
We establish the fact that in a Boolean network, the large deviation
principles of the family of laws
$P\left[(L^{0}_{1},L^{0}_{2})\in\Gamma\right]$ with speed $\lambda$ and rate
function $I(w,\pi)$ is given by

$\displaystyle-\inf_{(w,\pi)\in\Gamma^{0}}I(w,\pi)$
$\displaystyle\leq\liminf_{\lambda\rightarrow\infty}\dfrac{1}{\lambda}\log
P\left[(L^{0}_{1},L^{0}_{2})\in\Gamma\right]\leq\limsup_{\lambda\rightarrow\infty}\dfrac{1}{\lambda}\log
P\left[(L^{0}_{1},L^{0}_{2})\in\Gamma\right]$
$\displaystyle\leq-\inf_{(w,\pi)\in cl(\Gamma)}I(w,\pi)$

During the research, we realized that there is a general category of
assumptions for which the probability of connection in our settings converges.
Different assumptions will lead to an LDP with a different rate. We may
encourage researchers to try various classes of assumptions on this model and
observe the various variations of the rate of the LDP.

Also, there is a penury of articles using the boolean model to describe the
happenings in telecommunication systems and study the interactions in such
systems. Researchers may take on the task of populating the literature with
state-of-the-art work using Boolean model in spatial telecommunication
modelling to be able to study the Gibbs measure in this setting.

Finally, based on the rate function obtained, a maximization problem can then
be solved involving the Gibbs distribution of this system to detect the best
configuration of the spatial telecommunication system setting used in this
work that minimizes the breach in communication.

Conflict of Interest

The authors declare that they have no conflict of interest.

## References

  * [1] Baccelli, F. and Blaszczyszyn, B.(2009).  Stochastic Geometry and Wireless Networks,  Vol. I,Theory Now Publishers Inc.
  * [2] Baccelli, F. and Blaszczyszyn, B.(2009).  Stochastic Geometry and Wireless Networks,  Vol. II,Applications Now Publishers Inc.
  * [3] Biggins, J. D. (2004).  Large deviations for mixtures  In Electron. Comm. Probab.9 60–71.
  * [4] Bangerter, B., Talwar, S., Arefi, R. and Stewart,K.(2014)  Networks and devices for the 5G era,  52(2),90-96.,IEEE.
  * [5] Doku-Amponsah, K.(2016).  Joint large deviation result for empirical measures of the coloured random geometric graphs,  SpringerPlus 5,1140.
  * [6] Doku-Amponsah, K.(2006).  Large deviations and basic information theory for hierarchical and networked data structures,  Ph.D Thesis, University of Bath (UK),ProQuest Dissertations Publishing,2006.
  * [7] Daley, D.J. and Vere-Jones,D.(2006).  I: Elementary Theory and Methods, Second Edition,Springer-Verlag, New York, ISBN : 978-0-387-21337-8.
  * [8] A. Dembo and O. Zeitouni.(1998)  Large deviations techniques and applications.  Springer, New York, 2nd ed. 1998.
  * [9] Haenggi, M.(2010).  Advanced Topics in Random Wireless Networks  Course notes.
  * [10] Hirsch, C., Jahnel, B. and Patterson, R.(2018).  Space-time large deviations in capacity-constrained relay networks. ALEA, Lat. Am. J. Probab. Math. Stat. 15, 587–615
  * [11] Hirsch, C.m, Jahnel, B., Keeler, P. and Patterson, R.(2016).  Large deviation principles for connectable receivers in wireless networks,  Advances in Applied Probability 48(4),1061-1094.
  * [12] Jahnel, B. and König,W. (2020).  Probabilstic Methods in Telecommunication.  Birkhäuser Basel, Compact Textbooks in Mathematics
  * [13] Matthew, A. T. and Kottas, A.(2012).  Mixture Modeling for Marked Poisson Processes,  Bayesian Analysis, 7(20,335-362.
  * [14] Nketia, J.H.K. Drumming in Akan Communities of Ghana,  Edinburgh,Published online by Cambridge University Press: 22 January 2009
  * [15] Locke, D.(1990)  Drum Damba: Talking Drum Lessons,  Crown Point,White Cliffs Media Co; First Edition (January 1, 1990)
  * [16] Robins, H.(1995).  A Remark on Stirling’s Formula  The American Mathematical Monthly 62(1), 26–29.
  * [17] Sakyi-Yeboah,E. and Doku-Amponsah,K. and Asiedu,L.(2020).  Local large deviation principle, large deviation principle and information theory for the signal-to- interference-plus-noise ratio graph models,  _Journal of Information and Optimization Sciences_ , 42:1, 249-273
  * [18] Sakyi-Yeboah, E. and Kwofie, C. and Asiedu, L. and Doku-Amponsah, K.(2021).  Large deviation principle for empirical SINR measure of critical telecommunication networks,  _Journal of Information and Optimization Sciences_ 42(1),287-301.
  * [19] Sakyi-Yeboah, E., Andam, P., Asiedu, L. and Doku-Amponsah, K.(2020).  Large Deviations, Sharron-MacMillan-Breiman Theorem for Super-Critical Telecommunication Networks  To appear _Journal of Information and Optimization Sciences_.
  * [21] Umar, I. , Lotis,A.and Doku-Amponsah,K(2021).  From Statistical Mechanics to Large Deviations of Uniformly Random D-Regular Graphs,  To appear _Journal of discrete Mathematics and Cythography_