# Large Deviations and Information theory for Sub-Critical for the Signal -to-
Interference -Plus- Noise Ratio Randon Network Models

By E. Sakyi-Yeboah1, P. S. Andam, L. Asiedu1 and K. Doku-Amponsah1,2

1Department of Statistics and Actuarial Science, University of Ghana, BOX LG
115, Legon,Accra

2 Email<EMAIL_ADDRESS>
2 Telephone: +233205164254

††Acknowledgement:  This Research work has been supported by funds from the
Carnegie Banga-Africa Project, University of Ghana

> Abstract. The article obtains large deviation asymptotic for sub-critical
> communication networks modelled as signal-interference-noise-ratio(SINR)
> random networks. To achieve this, we define the empirical power measure and
> the empirical connectivity measure, as well as prove joint large deviation
> principles(LDPs) for the two empirical measures on two different scales.
> Using the joint LDPs, we prove an Asymptotic equipartition property(AEP) for
> wireless telecommunication Networks modelled as the subcritical SINR random
> networks. Further, we prove a Local Large deviation principle(LLDP) for the
> sub-critical SINR random network. From the LLDPs, we prove the large
> deviation principle, and a classical McMillan Theorem for the stochastic
> SINR model processes. Note that, the LDPs for the empirical measures of this
> stochastic SINR random network model were derived on spaces of measures
> equipped with the $\tau-$ topology, and the LLDPs were deduced in the space
> of SINR model process without any topological limitations. We motivate the
> study by describing a possible anomaly detection test for SINR random
> networks.

Keywords:  Large deviation principle, Sub-critical SINR random network model,
Poisson point process, Empirical power measure, Empirical connectivity
measure. Relative entropy,Kullback action,Spectral potential, Anomaly
detection test, Cybersecurity.

AMS Subject Classification: 60F10, 05C80, 68Q87, 28D20

##  1\. Introduction

In telecommunication, Wireless networks are usually modelled by the SINR
random networks. In the SINR random network model two nodes are deemed to
communicate if SINR is bigger than a certain threshold as specified by some
technical constant. In the process of addressing the additional requirement
imposed on wireless communication networks, in particular, a higher
availability of a highly accurate modeling of the SINR is required. Example,
each transmission may be equipped with some battery power which may be called
the mark of the node and the quantity SINR defined by the inclusion of the
marks in the definition. Further study of the SINR network model has shown
that an SINR model of interference is a more realistic model of interference
than the protocol model of interference: a receiver node receives a packet so
long as the signal to interference plus noise ratio is above a certain
threshold. See, Bakshi et al. [2].

There are many applications of large deviation techniques to the SINR
networks, which are used as models for telecommunication networks. Some of
these applications include, the analysis of bi-stability in networks, such as
notorious bi-stability in multiple access protocols the Aloha, and the
stochastic behaviour of ATM the admission control, sizing of internal buffers,
and the simulation of ATM models, see,[13]. and prevention of cyber-attacks on
wireless telecommunication networks, see example [12].

Cybersecurity of the devices in a telecommunication system is a major issue
when the devices become increasing dependent on computer and other local
networks. And an anomaly detection in the devices networks is key to avoiding
disruption in the telecommunication systems. Cybersecurity of the intelligent
electronic devices in telecommunication substations has been recognized as a
critical issue for smooth running of the system. One main approach to dealing
with these issues is to develop new technologies to detect and disrupt any
malicious activities over the networks.An Anomaly detection may be regarded as
an early warning mechanism to extract relevant cybersecurity events from
devices locations and correlate these events. Large deviation principles have
played key role in the formulation of efficient anomaly inference algorithm
for systems such as power grid, Wireless Sensor Network systems and
Telecommunication systems.

In this article, we prove joint large deviation principles on the scales
$\lambda$ and $\lambda^{2}a_{\lambda}$, where $\lambda$ is the intensity
measure of the underlining PPP of the subcritical SINR model. See, [9] or [10]
or [11] for similar results fore the dense SINR random network models. From
these LDPs, we prove an asymptotic equipartition property; see example [9],
for the SINR models.

Further, the study shows a LLDP for the SINR models. See example, [9] and
references therein. From the LLDP, we deduce asymptotic bounds on the
cardinality of the set of SINR models for a given typical empirical marked
measure. In addition, the study shows that from the LLDP an LDP for the SINR
modelled processes.

###  1.1. Background

This study set a dimension $d\in\mathbb{N}$ and some measureable set
${\mathcal{D}}\subset\mathbb{R}^{k}$ with reference to the Borel-$\sigma$
algebra ${\mathcal{B}}(\mathbb{R}^{d}).$ Given
$\lambda\mu:{\mathcal{D}}\to[0,1]$, an intensity measure and probability
kernel density function from ${\mathcal{D}}$ to $\mathbb{R}^{+}$,
${\mathcal{K}}$ and a path loss model, $\pi(\eta)=\eta^{-\alpha},$ where
$\alpha\in\mathbb{R}^{+},$ and some technical constraint;
$\iota^{(\lambda)},\zeta^{(\lambda)}:\mathbb{R}^{+}\to\mathbb{R}^{+}.$ The
study defined the SINR network model to as follows:

  * •

We select $\sigma=(\sigma_{u})_{u\in I},$ a Poisson Point Process (PPP) with
rate measure $\lambda\mu:{\mathcal{D}}\to[0,1]$.

  * •

Given the process $\sigma,$ the locations, each $\sigma_{u}$ is assigned a
mark or power $\ell(\sigma_{u})=\ell_{u}$ independently according to the
kernel density function ${\mathcal{K}}(\cdot\,,\,\sigma_{u}).$

  * •

For any two set of marked points
$((\sigma_{u},\ell_{u}),(\sigma_{v},\ell_{x}))$ we link an edge if and only if

$SINR(\sigma_{u},\sigma_{v},\sigma)\geq\iota^{(\lambda)}(\ell_{v})\mbox{ and
$SINR(\sigma_{v},\sigma_{u},\sigma)\geq\iota^{(\lambda)}(\ell_{u}),$}$

where

$SINR(\sigma_{v},\sigma_{u},\sigma)=\frac{\ell_{u}\pi(\|\sigma_{u}-\sigma_{v}\|)}{N_{0}+\zeta^{(\lambda)}(\ell_{v})\sum_{u\in
I\setminus\\{v\\}}\ell_{u}\pi(\|\sigma_{u}-\sigma_{v}\|)}$

We let $E$ denote the set of edges in the SINR random network and observe
$Y^{\lambda}:=Y^{\lambda}(\ell,\sigma,\mu)=\Big{\\{}[(\sigma_{u},\ell_{u}),u\in
I],\,E\Big{\\}}$ under the joint law of the marked PPP and the network. In
this article, we call $Y^{\lambda}$ an SINR Network model and
$(\sigma_{u},\ell_{u}):=\sigma_{u}^{\lambda}$ as the mark of site $u.$ Recall
from [9] that if $N_{0}=0$, then the connectivity function of the SINR random
network model, $T^{\lambda}$, is defined as
$T^{\lambda}((u,\ell_{u}),(v,\ell_{v}))=e^{-\lambda
t_{\lambda}^{{\mathcal{D}}}((u,\ell_{u}),(v,\ell_{v}))},$ where

$t_{\lambda}^{{\mathcal{D}}}((u,\ell_{u}),(v,\ell_{v}))=\int_{D}\Big{[}\mbox{$\frac{\iota^{(\lambda)}(\ell_{u})\zeta^{(\lambda)}(\ell_{u})}{\iota^{(\lambda)}(\ell_{u})\zeta^{(\lambda)}(\ell_{u})+(\|r\|^{\eta}/\|u-v\|^{\eta})}$}+\mbox{$\frac{\iota^{(\lambda)}(\ell_{v})\zeta^{(\lambda)}(\ell_{v})}{\iota^{(\lambda)}(\ell_{v})\zeta^{(\lambda)}(\ell_{v})+(\|r\|^{\eta}/\|v-u\|^{\eta})}$}\Big{]}\mu(dr).$

This article assumes that there exists $a_{\lambda}$ and a function
$t:{\mathcal{D}}\times\mathbb{R}_{+}\to(0,\infty)$ such that
$\lambda^{2}a_{\lambda}\to 0$ and

$\displaystyle\lim_{\lambda\uparrow\infty}a_{\lambda}^{-1}T^{\lambda}((a,\ell_{a}),(b,\ell_{b}))=t((a,\ell_{a}),(b,\ell_{b})).$

Sakyi-Yeboah et. al [10] and Sakyi-Yeboah et. al [11] investigates the
critical SINR network model (that is $\lambda a_{\lambda}\to 1$) and super-
critical SINR network model ( that is $\lambda a_{\lambda}\to\infty$)
respectively . In this articles, we shall focus this study on sub-critical
SINR Networks( that is $\lim_{\lambda\to\infty}\lambda a_{\lambda}\to 0$).

For a given set ${\mathcal{D}})$ we define ${\mathcal{S}}({\mathcal{D}})$ by

${\mathcal{S}}({\mathcal{D}})=\cup_{x\subset{\mathcal{D}}}\Big{\\{}x:\,\,|x\cap
W|<\infty\,\,,\mbox{for\, any bounded $W\subset{\mathcal{D}}$ }\Big{\\}}.$
(1.1)

Let ${\mathcal{W}}={\mathcal{S}}({\mathcal{D}}\times\mathbb{R}_{+})$ and
${\mathcal{M}}({\mathcal{W}})$, represent the space of positive measures on
the space ${\mathcal{W}}$ equipped with $\tau-$ topology. Note,
${\mathcal{W}}$ is a locally finite subset of the set
${\mathcal{D}}\times\mathbb{R}_{+}.$ See, example, [10]. Without abuse of
notation we shall refer to ${\mathcal{M}}({\mathcal{W}}\times{\mathcal{W}})$
as the space of symmetric measure on ${\mathcal{W}}\times{\mathcal{W}}$
endowed with the $\tau-$ topology. For any SINR random network model
$Y^{\lambda}$ we define a probability measure, the _empirical power measure_ ,
$M_{1}^{Y^{\lambda}}\in{\mathcal{M}}({\mathcal{W}})$, by

$M_{1}^{Y^{\lambda}}\big{(}(a,\ell_{a})\big{)}:=\frac{1}{\lambda}\sum_{u\in{\mathcal{W}}}\delta_{\sigma_{u}^{\lambda}}\big{(}(a,\ell_{a})\big{)}$

and a finite measure, the _empirical connectivity measure_
$M_{2}^{Y^{\lambda}}\in{\mathcal{M}}({\mathcal{W}}\times{\mathcal{W}}),$ by

$M_{2}^{Y^{\lambda}}\big{(}(a,\ell_{a}),(b,\ell_{b})\big{)}:=\frac{1}{\lambda^{2}a_{\lambda}}\sum_{(u,v)\in
E}[\delta_{(\sigma_{u}^{\lambda},\sigma_{v}^{\lambda})}+\delta_{(\sigma_{v}^{\lambda},\sigma_{u}^{\lambda})}]\big{(}(a,\ell_{a}),(b,\ell_{b})\big{)}.$

It should be noted that the total mass $\|M_{1}^{Y^{\lambda}}\|$ of the
empirical power measure is
${\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}$
and total mass of the empirical connect measure is
$2|E|/\lambda^{2}a_{\lambda}$.

###  1.2. Motivation: Anomaly detection in spatial networks

Consider, SINR random network model as a model that account for the
connectivity structure of the Wireless telecommunication networks (WTN). In
particular, consider the subcritical SINR random networks as model for the
WTNs since, in the implementation, the multihop network formed by the sensor
nodes may adopt a network structure. The network will be formed randomly
according to an arbitrary rule that is dependent on the distances between the
device locations. Assume the device locations are marked according to their
battery power, and the propagation of events is un-directed on the network.
Our objective is to estimate network parameters and possible identify possible
deviations form the actual values.

For instance, given a long sequence of realization $Y^{\lambda,k}$ of this
sub-critical marked SINR random network, one would like to approximate
parameter of the model, $\mu\times{\mathcal{K}}$ and $t$, by taking the
average frequencies of the corresponding samples. In particular, if
$M_{1}^{Y^{\lambda,k}}$ and $M_{2}^{Y^{\lambda,k}}$; the empirical power
measure and the empirical connectivity measure of $Y^{\lambda}$, the $k^{th}$
realization then

$\displaystyle\lim_{k\to\infty}\mbox{$\frac{1}{k}$}\sum_{r=1}^{k}M_{1}^{Y^{\lambda,r}}(a,\ell_{a})\to\mu\otimes{\mathcal{K}}(a,\,\ell_{a})$

and

$\lim_{k\to\infty}\Big{[}\mbox{$\frac{1}{k}$}\sum_{r=1}^{k}M_{2}^{Y^{\lambda,r}}\Big{(}(a,\ell_{a}),(b,\ell_{b})\Big{)}/\mbox{$\frac{1}{k}$}\sum_{r=1}^{k}M_{2}^{Y^{\lambda,k}}(a,\ell_{a})\otimes\mbox{$\frac{1}{k}$}\sum_{r=1}^{k}M_{1}^{Y^{\lambda,r}}(b,\ell_{b})\Big{]}\to
t\Big{(}(a,\ell_{a}),(b,\ell_{b})\Big{)},$

with probability $1.$

Assuming that we have estimated $\mu\otimes{\mathcal{K}}$ and $t.$ We are
interested in a test that determines whether a particular realization
$Y^{\lambda}$ is typical or not. Thus, we want to differentiate between
$\mu\times{\mathcal{K}}$ and $t$ (Hypothesis $H_{0}$) and any other unknown
law (Hypothesis $H_{1}$). Theorem 2.1 will be the bases of providing
generalized Neyman-Pearson criterion,See [7, pp.96-100],and hence an anomaly
detection test for the sub-critical marked SINR random networks.

This article is structured as follows: Section 2 presents the main results;
Theorem 2.1, Theorem 2.2, Theorem 2.3, Corollary 2.4 and Corollary 2.5. In
Section 3 we prove the main results of the article, Theorem 2.1. Section 4
provides the proof of the AEP, see Theorem 2.2 and Section 5; Proof of Theorem
2.3, Corollary 2.4 and Corollary 2.5. Lastly, Section 6 presents the
conclusion to the article.

##  2\.  Main Results

Theorem 2.1, is a joint large deviation principle for the empirical measures
of the SINR network models.With reference from Subsection 1.1, we recall the
definition of $t_{\lambda}^{{\mathcal{D}}}$ as

$t_{\lambda}^{{\mathcal{D}}}((a,\ell_{a}),(b,\ell_{b}))=\int_{D}\Big{[}\mbox{$\frac{\iota^{(\lambda)}(\ell_{u})\zeta^{(\lambda)}(\ell_{u})}{\iota(\ell_{u})\zeta(\ell_{u})+(\|r\|^{\eta}/\|i-y\|^{\eta})}$}+\mbox{$\frac{\tau^{(\lambda)}(\ell_{v})\gamma^{(\lambda)}(\ell_{v})}{\tau^{(\lambda)}(\ell_{v})\gamma^{(\lambda)}(\ell_{v})+(\|r\|^{\eta}/\|y-x\|^{\eta})}$}\Big{]}\mu(dr)$

and note that

$t\beta\otimes\beta((a,\ell_{a}),(b,\ell_{b}))):=t((a,\ell_{a}),(b,\ell_{b}))\mu((a,\ell_{a}))\mu((b,\ell_{b})).$

######  Theorem 2.1.

Let $Y^{\lambda}$ is a sub-critical marked SINR network model with rate
measure $\lambda\mu:{\mathcal{D}}\to[0,1]$ and a power transition kernel
function ${\mathcal{K}}(\cdot,y)=ce^{-cy},y>0$ and path loss function
$\pi(\eta)=\eta^{-\alpha},$ for $\alpha>0.$ Thus, the link kernel function
$T^{\lambda}$ of $Y^{\lambda}$ satisfies $a_{\lambda}^{-1}T^{\lambda}\to t$
and $\lambda a_{\lambda}\to 0.$ Then, as $\lambda\to\infty$, the pair of
measures $(M_{1}^{Y^{\lambda}},M_{2}^{Y^{\lambda}})$ satisfies a large
deviation principle in the space
${\mathcal{M}}({\mathcal{W}})\times{\mathcal{M}}({\mathcal{W}}\times{\mathcal{W}})$

  * (i)

with speed $\lambda$ and a good rate function

$\displaystyle
I^{1}\big{(}\beta,\phi\big{)}=\left\\{\begin{array}[]{ll}H\Big{(}\beta\Big{|}\mu\otimes{\mathcal{K}}\Big{)}&\,\,\mbox{
if $\phi=t\beta\otimes\beta$ }\\\ \infty&\mbox{elsewhere.}\end{array}\right.$
(2.1)

  * (ii)

with speed $\lambda^{2}a_{\lambda}$ and good rate function

$\displaystyle
I^{2}\big{(}\beta,\phi\big{)}=\left\\{\begin{array}[]{ll}{\mathcal{H}}(\phi\|t\beta\otimes\beta),&\,\,\mbox{
if $\beta=\mu\otimes{\mathcal{K}}$ }\\\
\infty&\mbox{elsewhere.}\end{array}\right.$ (2.2)

where

$\displaystyle{\mathcal{H}}(\phi\|t\beta\otimes\beta):=\left\\{\begin{array}[]{ll}H(\phi\,\|\,t\beta\otimes\beta)+\Big{(}\|t\beta\otimes\beta\|-\|\phi\|\Big{)},&\mbox{if
$\|\phi\|>0.$ }\\\ \infty&\mbox{elsewhere.}\end{array}\right.$ (2.3)

######  Theorem 2.2.

Suppose $Y^{\lambda}$ be a sub-critical marked SINR network model with rate
measure $\lambda\beta:D\to[0,1]$ and a power probability function
${\mathcal{K}}(\cdot,y)=ce^{-cy},y>0$ and path loss function
$\pi(\eta)=\eta^{-\alpha},$ for $\alpha>0.$ Thus, the connectivity probability
$T^{\lambda}$ of $Y^{\lambda}$ satisfies $a_{\lambda}^{-1}T^{\lambda}\to t$
and $\lambda a_{\lambda}\to 0.$ Suppose the sequence $a_{\lambda}$ of
$Y^{\lambda}$ is such that $\lambda a_{\lambda}\,\log\lambda\to 0$ and
$a_{\lambda}/\log\lambda\to-1.$ Then, we have

$\lim_{\lambda\to\infty}\mathbb{P}\Big{\\{}\Big{|}-\frac{1}{a_{\lambda}\lambda^{2}\log\lambda}\log
P(Y^{\lambda})-\mathbb{E}_{f}\Big{[}t((\cdot,\cdot),(\cdot,\cdot))\Big{]}\Big{|}\geq\varepsilon\Big{\\}}=0,$

where the expectation was taken with respect to the distribution function

$f((x,\ell_{x}),(y,\ell_{y}))=c^{2}e^{-c(x+y)}\mu(d\ell_{x})\mu(d\ell_{y})dxdy,\,\mbox{
$x>0,y>0,\ell_{x}>0,\ell_{y}>0.$}$

Note that the
$H(f):=\mathbb{E}_{f}\Big{[}t((\cdot,\cdot),(\cdot,\cdot))\Big{]}$ is an
entropy.

Interpretation: To transmit information contain in a large SINR random network
modls one require with a large probability

$-\lambda^{2}a_{\lambda}\log\lambda\Big{[}H(f)\Big{]}/\log 2\mbox{ \bf bits.}$

Let ${\mathcal{G}}$ be the set of all SINR networks with rate measure
$\lambda\mu:{\mathcal{D}}\to[0,1]$ and state the Local Large deviation
principle as follows:

######  Theorem 2.3.

Suppose $Y^{\lambda}$ is a sub-critical marked SINR network model with rate
measure $\lambda\mu:{\mathcal{D}}\to[0,1]$ and a mark transition kernel
${\mathcal{M}}(y)=c{e}^{-cy},y>0$ and path loss function
$\pi(\eta)=\eta^{-\alpha},$ for $\eta>0$ and $\alpha>0.$ Thus, the link
probability $T^{\lambda}$ of $Y^{\lambda}$ satisfies
$a_{\lambda}^{-1}T^{\lambda}\to t$ and $\lambda a_{\lambda}\to 0.$ Then,

  * •

for any functional $\phi\in{\mathcal{G}}$ and a number $\varepsilon>0$, there
exists a weak neighbourhood $B_{\phi}$ such that

$\mathbb{P}_{\beta}\Big{\\{}Y^{\lambda}\in{\mathcal{G}}\,\Big{|}\,M_{2}^{Y^{\lambda}}\in
B_{\phi}\Big{\\}}\leq
e^{-\mbox{$\frac{1}{2}$}\lambda^{2}a_{\lambda}{\mathcal{H}}(\phi\|t\beta\otimes\beta)-\lambda
a_{\lambda}\varepsilon},\,\mbox{where $\beta=\mu\otimes{\mathcal{K}}.$ }$

  * •

for any $\phi\in{\mathcal{G}}_{\beta}$, a number $\varepsilon>o$ and a fine
neighbourhood $B_{\phi}$, we have the compute:

$\mathbb{P}_{\beta}\Big{\\{}Y^{\lambda}\in{\mathcal{G}}\,\Big{|}\,M_{2}^{Y^{\lambda}}\in
B_{\phi}\Big{\\}}\geq
e^{-\mbox{$\frac{1}{2}$}\lambda^{2}a_{\lambda}{\mathcal{H}}(\phi\|t\beta\otimes\beta)+\lambda_{\lambda}a_{\lambda}\varepsilon},\,\mbox{where
$\beta=\mu\otimes{\mathcal{K}}.$ }$

For the given telecommunication network model, we define an entropy as
$h:{\mathcal{M}}({\mathcal{W}}\times{\mathcal{W}})\to[0.\infty]$ by

$h(\phi):=\Big{(}\|\phi\|-\|\lambda\beta\otimes\beta\|-\Big{\langle}\phi\,,\,\log\mbox{$\frac{\phi}{\|t\beta\otimes\beta\|}$}\Big{\rangle}\Big{)}/2,\,\,\mbox{
where $\beta=\mu\otimes{\mathcal{K}}.$}$ (2.4)

######  Corollary 2.4 (McMillian Theorem).

Let $Y^{\lambda}$ be a sub-critical marked SINR network model with rate
measure $\lambda\mu:{\mathcal{D}}\to[0,1]$ and a mark transition kernel
${\mathcal{K}}(\cdot,\,y)=c{e}^{-cy},y>0$ and path loss function
$\pi(\eta)=\eta^{-\alpha},$ for $\eta>0$ and where
$\beta=\mu\otimes{\mathcal{K}}.$ $\alpha>0.$ Thus, the link probability
$T^{\lambda}$ of every $Y^{\lambda}\in{\mathcal{G}}$ satisfies
$a_{\lambda}^{-1}T^{\lambda}\to t$ and $\lambda a_{\lambda}\to 0.$

  * (u)

For any empirical link measure $\phi$ on ${\mathcal{W}}\times{\mathcal{W}}$
and $\varepsilon>0,$ there exists a neighborhood $B_{\phi}$ such that

$Card\Big{(}\big{\\{}Y^{\lambda}\in{\mathcal{G}}\,|\,M_{2}^{Y^{\lambda}}\in
D_{\phi}\big{\\}}\Big{)}\geq
e^{\lambda^{2}a_{\lambda}(h(\phi)-\varepsilon\big{)}}.$

  * (ii)

for any neighborhood $B_{\rho}$ and $\varepsilon>0,$ we have

$Card\Big{(}\big{\\{}Y^{\lambda}\in{\mathcal{G}}\,|\,M_{2}^{Y^{\lambda}}\in
B_{\phi}\big{\\}}\Big{)}\leq
e^{\lambda^{2}a_{\lambda}(h(\phi)+\varepsilon\big{)}},$

where $Card(\gamma)$ means the cardinality of $\gamma.$

###### remark 1

Given $\phi=t\beta\otimes\beta,$ we have $\displaystyle
Card\Big{(}\Big{\\{}j\in{\mathcal{G}}\,\Big{\\}}\Big{)}\approx
e^{\lambda^{2}\,a_{\lambda}\|t\beta\otimes\beta\|{\mathcal{H}}\big{(}t\beta\otimes\beta/\|t\beta\otimes\beta\|\big{)}},$
where $\beta=\mu\otimes{\mathcal{K}}.$

Interpretation: Note from Corollary 2.4 that, for the typical empirical
connectivity measure, $t\mu^{2}\otimes{\mathcal{K}}^{2},$ the cardinality of
the space of SINR models is nearly equal to $\displaystyle
e^{\lambda^{2}a_{\lambda}\|t\mu^{2}\otimes{\mathcal{K}}^{2}\|H\big{(}t\mu^{2}\otimes{\mathcal{K}}^{2}/\|t\mu^{2}\otimes{\mathcal{K}}^{2}\|\big{)}}.$
The next theorem is the LDP for the SINR random network processes.

######  Corollary 2.5.

Let $Y^{\lambda}$ be a sub-critical marked SINR random network model with rate
measure $\lambda\mu:{\mathcal{D}}\to[0,1]$ and a mark kernel function
${\mathcal{K}}(\cdot,\,y)=c{e}^{-cy},y>0$ and path loss function
$\pi(\eta)=\eta^{-\alpha},$ for $\alpha>0.$ Thus, the link probability
$T^{\lambda}$ of $Y^{\lambda}$ satisfies $a_{\lambda}^{-1}T^{\lambda}\to t$
and $\lambda a_{\lambda}\to 0.$

  * •

Let $U$ be closed subset ${\mathcal{G}}$. Then we have

$\limsup_{\lambda\to\infty}\frac{1}{\lambda^{2}a_{\lambda}}\log\mathbb{P}_{\mu\times{\mathcal{K}}}\Big{\\{}Y^{\lambda}\in{\mathcal{G}}\,\Big{|}\,M_{2}^{Y^{\lambda}}\in
U\Big{\\}}\leq-\mbox{$\frac{1}{2}$}\inf_{\phi\in
U}\Big{\\{}{\mathcal{H}}(\phi\|t\mu\times{\mathcal{K}}\otimes\mu\times{\mathcal{K}})\Big{\\}}$

  * •

Let $O$ be open subset ${\mathcal{G}}$. Then we have

$\liminf_{\lambda\to\infty}\frac{1}{\lambda^{2}a_{\lambda}}\log\mathbb{P}_{\beta}\Big{\\{}Y^{\lambda}\in{\mathcal{G}}\,\Big{|}\,M_{2}^{Y^{\lambda}}\in
O\Big{\\}}\geq-\mbox{$\frac{1}{2}$}\inf_{\phi\in
O}\Big{\\{}{\mathcal{H}}(\phi\|t\mu\times{\mathcal{K}}\otimes\mu\times{\mathcal{K}})\Big{\\}}.$

##  3\. Proof of Main Results

###  3.1. Proof of Theorem 2.1(i)

Suppose $W_{1},...,W_{n}$ is a decomposition of the space
${\mathcal{D}}\times\mathbb{R}_{+}.$ Note that, for every $(u,v)\in
A_{x}\times A_{y},\,x,y=1,2,3,...,n,$ $\lambda M_{2}^{Y^{\lambda}}(u,v)$ given
$\lambda M_{1}^{Y^{\lambda}}(u)=\lambda\beta(u)$ denotes a number of bernoulli
trial with parameters $\lambda^{2}\beta(u)\beta(v)/2$ and $T^{\lambda}(u,v).$
Consider ${\mathcal{K}}$ to represent as the gamma distribution with mean
$1/c.$ With reference to the function $t_{\lambda}^{{\mathcal{D}}}$ from the
preceding sections, we observe that Lemma 2.3 is fundamental in the
application of the Gartner-Ellis Theorem. See [7].

######  Lemma 3.1.

Suppose $Y^{\lambda}$ is a sub-critical marked SINR random model with rate
measure $\lambda\mu:{\mathcal{D}}\to[0,1]$ and a power probability function
${\mathcal{K}}(\cdot,\,y)=c{e}^{-cy},y>0$ and path loss function
$\pi(\eta)=\eta^{-\alpha},$ for $\eta>0$ and $\alpha>0.$ Thus, the link
probability $T^{\lambda}$ of $Y^{\lambda}$ satisfies
$a_{\lambda}^{-1}T^{\lambda}tot$ and $\lambda a_{\lambda}\to 0.$ Suppose
$Y^{\lambda}$ be a sub-critical SINR network model, conditional on the event
$M_{1}^{Y^{\lambda}}=\beta.$ Let
$q:{\mathcal{W}}\times{\mathcal{W}}\to\mathbb{R}$ be bounded function. Then,

$\displaystyle\lim_{\lambda\to\infty}\frac{1}{\lambda}\log\mathbb{E}\Big{\\{}e^{\lambda\langle
q,\,M_{2}^{Y^{\lambda}}\rangle}\Big{|}M_{1}^{Y^{\lambda}}=\beta\Big{\\}}$
$\displaystyle=\frac{1}{2}\lim_{n\to\infty}\sum_{y=1}^{n}\sum_{x=1}^{n}\Big{\langle}q,\,t\beta\otimes\beta\Big{\rangle}_{A_{x}\times
A_{y}}$
$\displaystyle=\frac{1}{2}\Big{\langle}q,\,t\beta\otimes\beta\Big{\rangle}_{{\mathcal{W}}\times{\mathcal{W}}}.$

Proof.  Now we observe that

$\mathbb{E}\Big{\\{}e^{\int\int\lambda
q(u,v)M_{2}^{Y^{\lambda}}(du,dv)/2}\Big{|}M_{1}^{Y^{\lambda}}=\beta\Big{\\}}=\mathbb{E}\Big{\\{}\prod_{u\in{\mathcal{W}}}\prod_{v\in{\mathcal{W}}}e^{\lambda
q(u,v)_{2}^{\lambda}(du,dv)/2}\Big{\\}}$

$\mathbb{E}\Big{\\{}\prod_{u\in{\mathcal{W}}}\prod_{v\in{\mathcal{W}}}e^{q(u,v)\lambda
M_{2}^{Y^{\lambda}}(du,dv/2)}\Big{\\}}=\prod_{x=1}^{n}\prod_{y=1}^{n}\prod_{u\in
W_{x}}\prod_{v\in W_{y}}\mathbb{E}\Big{\\{}e^{q(u,v)\lambda
M_{2}^{Y^{\lambda}}(du,dv)/2}\Big{\\}}$

$\log\Big{\\{}e^{\lambda\langle
q,M_{2}^{Y^{\lambda}}\rangle/2}\Big{|}M_{1}^{Y^{\lambda}}=\beta\Big{\\}}=\sum_{y=1}^{n}\sum_{x=1}^{n}\int_{W_{y}}\int_{W_{x}}\log\Big{[}1-T^{\lambda}(u,v)+T^{\lambda}(u,v)e^{q(u,v)/\lambda
a_{\lambda}}\Big{]}^{\lambda^{2}\beta\otimes\beta(du,dv)/2}+o(n)$

Introducing the dominated convergence theorem

$\frac{1}{\lambda}\log E\\{e^{\lambda\langle
q,M_{2}^{Y^{\lambda}}\rangle/2}\mid
M_{1}^{Y^{\lambda}}=\beta\\}=\frac{1}{\lambda}\sum_{y=1}^{n}\sum_{x=1}^{n}\int_{W_{x}}\int_{W_{y}}\log\Big{[}1-\big{(}1-e^{q(u,v)/\lambda
a_{\lambda}})T^{\lambda}(u,v)\Big{]}^{\lambda^{2}\beta\otimes\beta(du,dv)/2}+o(n)/\lambda$
$\frac{1}{\lambda}\log\mathbb{E}\\{e^{\lambda\langle
q,M_{2}^{Y^{\lambda}}\rangle/2}\big{|}M_{1}^{Y^{\lambda}}=\beta\\}=\lim_{\lambda\rightarrow\infty}\sum_{y=1}^{n}\sum_{x=1}^{n}\int_{W_{x}}\int_{W_{y}}\log\Big{[}1+q(u,v)t(u,v)/\lambda+o(\lambda)/\lambda\Big{]}^{\lambda\beta\otimes\beta(du,dv)/2}+o(n)/\lambda$

$\lim_{\lambda\rightarrow\infty}\frac{1}{\lambda}\log\mathbb{E}\\{e^{\lambda\langle
q,M_{2}^{Y^{\lambda}}\rangle/2}\mid
M_{1}^{Y^{\lambda}}=\beta\\}=\frac{1}{2}\sum_{y=1}^{n}\sum_{x=1}^{n}\Big{\langle}g,\,t\beta\otimes\beta\Big{\rangle}_{W_{x}\times
W_{y}}$

$\displaystyle\lim_{\lambda\rightarrow\infty}\frac{1}{\lambda}\log\mathbb{E}\\{e^{\lambda\langle
q,M_{2}^{Y^{\lambda}}\rangle/2}\Big{|}M_{1}^{Y^{\lambda}}=\beta\\}$
$\displaystyle=\frac{1}{2}\lim_{n\rightarrow\infty}\sum_{y=1}^{n}\sum_{x=1}^{n}\Big{\langle}q,\,t\beta\otimes\beta\Big{\rangle}_{W_{x}\times
W_{y}}$
$\displaystyle=\frac{1}{2}\Big{\langle}q,\,t\beta\otimes\beta\Big{\rangle}_{{\mathcal{W}}\times{\mathcal{W}}}.$

Hence,by the Gartner-Ellis theorem, conditional on the event
$\Big{\\{}M_{1}^{Y^{\lambda}}=\beta\Big{\\}}$, $M_{2}^{Y^{\lambda}}$ obey a
large deviation principle with speed $\lambda$ and variational formulation of
the rate function

$I_{\beta}(\phi)=\frac{1}{2}\sup_{q}\Big{\\{}\Big{\langle}q,\,\phi\Big{\rangle}_{{\mathcal{W}}\times{\mathcal{W}}}-\Big{\langle}q,\,t\beta\otimes\beta\Big{\rangle}_{{\mathcal{W}}\times{\mathcal{W}}}\Big{\\}}$

the solution can be found, see example [4], would obviously reduces to the
good rate function as such

$I_{\beta}(\phi)=0.$ (3.1)

###  3.2. Proof of Theorem 2.1(ii)

Analogously we consider $W_{1},...,W_{n}$ as decomposition of the space
${\mathcal{D}}\times\mathbb{R}_{+}.$ We refer to $t_{\lambda}^{D}$ and observe
that, Lemma 3.2 will play an important role in the application of the Gartner-
Ellis Theorem. See, [7].

######  Lemma 3.2.

Let $Y^{\lambda}$ be a sub critical powered SINR network with rate measure
$\lambda\mu:{\mathcal{D}}\to[0,1]$ and a power probability function
${\mathcal{K}}(y)=c{e}^{-cy},y>0$ and path loss function
$\pi(\eta)=r^{-\alpha},$ for $\eta>0$ and $\alpha>0.$ Thus, the link
probability $T^{\lambda}$ of $Y^{\lambda}$ satisfies
$a_{\lambda}^{-1}T^{\lambda}\to t$ and $\lambda a_{\lambda}\to\infty.$ Let
$Y^{\lambda}$ be a sub-critical SINR network, conditional on the event
$M_{1}^{Y^{\lambda}}=\beta.$ Let
$q:{\mathcal{W}}\times{\mathcal{W}}\to\mathbb{R}$ be bounded function. Then,

$\displaystyle\lim_{\lambda\to\infty}\frac{1}{\lambda^{2}a_{\lambda}}\log\mathbb{E}\Big{\\{}e^{\lambda^{2}a_{\lambda}\langle
q,\,M_{2}^{Y^{\lambda}}\rangle}\Big{|}M_{1}^{Y^{\lambda}}=\beta\Big{\\}}$
$\displaystyle=-\frac{1}{2}\lim_{n\to\infty}\sum_{y=1}^{n}\sum_{x=1}^{n}\Big{\langle}1-e^{q},\,t\beta\otimes\beta\Big{\rangle}_{W_{x}\times
W_{y}}$
$\displaystyle=-\frac{1}{2}\Big{\langle}1-e^{q},\,t\beta\otimes\beta\Big{\rangle}_{{\mathcal{W}}\times{\mathcal{W}}}.$

Proof.  Now we note that

$\mathbb{E}\Big{\\{}e^{\int\int\lambda^{2}a_{\lambda}q(u,v)M_{2}^{Y^{\lambda}}(du,dv)/2}\Big{|}M_{1}^{Y^{\lambda}}=\beta\Big{\\}}=\mathbb{E}\Big{\\{}\prod_{i\in{\mathcal{W}}}\prod_{j\in{\mathcal{W}}}e^{\lambda^{2}a_{\lambda}q(u,v)M_{2}^{Y^{\lambda}}(du,dv)/2}\Big{\\}}$

$\mathbb{E}\Big{\\{}\prod_{i\in{\mathcal{W}}}\prod_{j\in{\mathcal{W}}}e^{q(u,v)\lambda
M_{2}^{Y^{\lambda}}(du,dv/2)}=\prod_{x=1}\prod_{y=1}\prod_{i\in
W_{x}}\prod_{j\in
W_{y}}\mathbb{E}\Big{\\{}e^{\lambda^{2}a_{\lambda}q(u,v)M_{2}^{Y^{\lambda}}(du,dv)/2}\Big{\\}}\times
e^{o(n)}$

$\log\Big{\\{}e^{\lambda^{2}a_{\lambda}\langle
q,M_{2}^{Y^{\lambda}}\rangle/2}\Big{|}M_{1}^{Y^{\lambda}}=\beta\Big{\\}}=\sum_{y=1}^{n}\sum_{x=1}^{n}\int_{W_{y}}\int_{W_{x}}\log\Big{[}1-T^{\lambda}(u,v))+T^{\lambda}(u,v)e^{q(u,v)}\Big{]}^{\lambda^{2}\beta\otimes\beta(du,dv)/2}+o(n)$

Using the dominated convergence theorem

$\frac{1}{\lambda^{2}a_{\lambda}}\log E\\{e^{\lambda\langle
q,M_{2}^{Y^{\lambda}}\rangle/2}\mid
M_{1}^{Y^{\lambda}}=\beta\\}=\frac{1}{\lambda^{2}a_{\lambda}}\sum_{y=1}\sum_{x=1}\int_{W_{x}}\int_{W_{y}}\log\Big{[}1-\big{(}1-e^{q(u,v)})T^{\lambda}(u,v)\Big{]}^{\lambda^{2}\beta\otimes\beta(du,dv)/2}+o(n)/\lambda^{2}a_{\lambda}$
$\frac{1}{\lambda^{2}a_{\lambda}}\log\mathbb{E}\\{e^{\lambda\langle
q,M_{2}^{Y^{\lambda}}\rangle/2}\big{|}M_{1}^{Y^{\lambda}}=\beta\\}=\lim_{\lambda\rightarrow\infty}\sum_{y=1}\sum_{x=1}\int_{W_{x}}\int_{W_{y}}\log\Big{[}1-(1-e^{q(u,v)})T^{\lambda}(u,v)\Big{]}^{\lambda\beta\otimes\beta(du,dv)/2}+o(n)/\lambda^{2}a_{\lambda}$

$\lim_{\lambda\rightarrow\infty}\frac{1}{\lambda^{2}a_{\lambda}}\log\mathbb{E}\Big{\\{}e^{\lambda\langle
q,M_{2}^{Y^{\lambda}}\rangle/2}\big{|}M_{1}^{Y^{\lambda}}=\beta\Big{\\}}=-\frac{1}{2}\sum_{y=1}\sum_{x=1}\int_{W_{x}}\int_{W_{y}}\Big{[}(1-e^{q(u,v)})t(u,v)\beta\otimes\beta(du,dv)\Big{]}$

$\lim_{\lambda\rightarrow\infty}\frac{1}{\lambda^{2}a_{\lambda}}\log\mathbb{E}\\{e^{\lambda\langle
q,M_{2}^{Y^{\lambda}}\rangle/2}\big{|}M_{1}^{Y^{\lambda}}=\beta\\}=-\frac{1}{2}\sum_{y=1}^{n}\sum_{x=1}^{n}\Big{\langle}1-e^{q},\,t\beta\otimes\beta\Big{\rangle}_{W_{x}\times
W_{y}}$

$\displaystyle\lim_{\lambda\rightarrow\infty}\frac{1}{\lambda^{2}a_{\lambda}}\log\mathbb{E}\\{e^{\lambda\langle
q,M_{2}^{Y^{\lambda}}\rangle/2}\Big{|}M_{1}^{Y^{\lambda}}=\beta\\}$
$\displaystyle=-\frac{1}{2}\lim_{n\rightarrow\infty}\sum_{y=1}^{n}\sum_{x=1}^{n}\Big{\langle}1-e^{q},\,t\beta\otimes\beta\Big{\rangle}_{W_{x}\times
W_{y}}$
$\displaystyle=-\frac{1}{2}\Big{\langle}1-e^{q},\,t\beta\otimes\beta\Big{\rangle}_{{\mathcal{W}}\times{\mathcal{W}}}$

Hence,by the Gartner-Ellis theorem, conditional on the event
$\Big{\\{}M_{1}^{Y^{\lambda}}=\beta\Big{\\}}$, $M_{2}^{Y^{\lambda}}$ obey a
large deviation principle with speed $\lambda$ and variational formulation of
the rate function is given by

$I_{\beta}(\phi)=\frac{1}{2}\sup_{q}\Big{\\{}\Big{\langle}q,\,\phi\Big{\rangle}_{{\mathcal{W}}\times{\mathcal{W}}}+\Big{\langle}1-e^{q},\,t\beta\otimes\beta\Big{\rangle}_{{\mathcal{W}}\times{\mathcal{W}}}\Big{\\}}$

which when solved, see example [4], will clearly reduce to the good rate
function given by

$I_{\beta}(\phi)=\frac{1}{2}{\mathcal{H}}(\phi\|t\beta\otimes\beta).$ (3.2)

###  3.3.  Proof of Theorem 2.1(ii) by Method of Mixtures.

For any $\lambda\in(0,\infty)$ we define

$\displaystyle{\mathcal{M}}_{\lambda}({\mathcal{W}})$
$\displaystyle:=\Big{\\{}\beta\in{\mathcal{M}}({\mathcal{W}})\,:\,\lambda\beta(u)\in\mathbb{N}\mbox{
for all }u\in{\mathcal{W}}\Big{\\}},$
$\displaystyle\tilde{\mathcal{M}}_{\lambda}({\mathcal{W}}\times{\mathcal{W}})$
$\displaystyle:=\Big{\\{}\phi\in{\mathcal{M}}({\mathcal{W}}\times{\mathcal{W}})\,:\,\lambda\,\phi(u,v)\in\mathbb{N},\,\mbox{
for all }\,u,v\in{\mathcal{W}}\Big{\\}}\,.$

We denote by $\Upsilon_{\lambda}:={\mathcal{M}}_{\lambda}({\mathcal{W}})$ and
$\Upsilon:={\mathcal{M}}({\mathcal{W}})$. We write

$\displaystyle P_{\beta_{\lambda}}^{(\lambda)}(\phi_{\lambda})$
$\displaystyle:=\mathbb{P}\big{\\{}M_{2}^{Y^{\lambda}}=\phi_{\lambda}\,\big{|}\,M_{1}^{Y^{\lambda}}=\beta_{\lambda}\big{\\}}\,,$
$\displaystyle P^{(\lambda)}(\beta_{\lambda})$
$\displaystyle:=\mathbb{P}\big{\\{}M_{1}^{Y^{\lambda}}=\beta_{\lambda}\big{\\}}$

Th joint distribution of $M_{1}^{Y^{\lambda}}$ and $M_{2}^{Y^{\lambda}}$ is
the mixture of $P_{\beta_{\lambda}}^{(\lambda)}$ with
$P^{(\lambda)}(\beta_{\lambda}),$ as follows:

$d\tilde{P}^{\lambda}(\beta_{\lambda},\ell_{\lambda}):=dP_{\beta_{n}}^{(\lambda)}(\ell_{\lambda})\,dP^{(\lambda)}(\beta_{\lambda}).\,$
(3.3)

(Biggins, Theorem 5(b), 2004) provides condition for the validity of large
deviation principles for the mixtures and for the goodness of the rate
function if individual large deviation principles are known. The following
three lemmas ensure validity of these conditions.

Note that the family of measures $({P}^{(\lambda)}\colon\lambda\in(0,\infty))$
is exponentially tight on $\Upsilon.$

######  Lemma 3.3.

  * (u)

The family of measures $(\tilde{P}^{\lambda}\colon\lambda\in(0,\infty))$ is
exponentially tight on
$\Upsilon\times\tilde{\mathcal{M}}({\mathcal{W}}\times{\mathcal{W}}).$

  * (ii)

The family measures $(T^{\lambda}\colon\lambda\in(0,\infty))$ is exponentially
tight on $\Upsilon\times{\mathcal{M}}({\mathcal{W}}\times{\mathcal{W}}).$

We refer to [9, Lemma 4.3] for similar proof for Large Deviation Principle on
the scale $\lambda^{2}$

Define the function
$I_{sc}^{2},I_{sc}^{1}\colon{\Upsilon}\times{\mathcal{M}}({\mathcal{W}}\times{\mathcal{W}})\rightarrow[0,\infty],$
by

$\displaystyle
I^{1}\big{(}\beta,\phi\big{)}=\left\\{\begin{array}[]{ll}H\Big{(}\beta\Big{|}\mu\otimes{\mathcal{K}}\Big{)}&\,\,\mbox{
if $\phi=t\beta\otimes\beta$ }\\\ \infty&\mbox{otherwise.}\end{array}\right.$
(3.4)

$I^{2}\big{(}\beta,\nu\big{)}=\frac{1}{2}{\mathcal{H}}\Big{(}\nu\|t\beta\otimes\beta\Big{)}.$
(3.5)

######  Lemma 3.4.

  * (u)

$I^{1}$ is lower semi-continuous.

  * (ii)

$I^{2}$ is lower semi-continuous.

By (Biggins, Theorem 5(b), 2004) the two previous lemmas, the LDP for the
empirical power measure, see, [9, Theorem 2.1] and the large deviation
principles we have established Theorem 2.1 ensure that under
$(\tilde{P}^{\lambda})$ and $T^{\lambda}$ the random variables
$(\beta_{\lambda},\ell_{\lambda})$ satisfy a large deviation principle on
${\mathcal{M}}({\mathcal{W}})\times{\mathcal{M}}({\mathcal{W}}\times{\mathcal{W}})$
and $\Upsilon\times{\mathcal{M}}_{\lambda}({\mathcal{W}}\times{\mathcal{W}})$
on the speeds $\lambda$ and $\lambda^{2}a_{\lambda}$ with good rate functions
$I^{1}$ and $I^{2}$ respectively, which ends the proof of Theorem 2.1.

##  4\. Proof of Theorem 2.2 by Large deviations

To prove the Shannon-Mcmillian Breiman (SMB) or the AEP, we first prove a weak
law of large numbers (WLLN) for the empirical marked measure and the empirical
connectivity measure of the SINR network model.

######  Lemma 4.1.

Let $Y^{\lambda}$ be a sub-critical marked SINR model with rate measure
$\lambda\mu:{\mathcal{D}}\to[0,1]$ and a marked transition function
${\mathcal{K}}(\cdot,y)=c{e}^{-cy},y>0$ and path loss function
$\pi(\eta)=\eta^{-\alpha},$ for $\alpha>0.$ Thus, the link probability
$T^{\lambda}$ of $Y^{\lambda}$ satisfies $a_{\lambda}^{-1}T^{\lambda}\to t$
and $\lambda a_{\lambda}\to 0.$ Then,

$\lim_{\lambda\to\infty}\mathbb{P}\Big{\\{}\sup_{(a,\ell_{a})\in{\mathcal{W}}}\Big{|}M_{1}^{Y^{\lambda}}(a,\ell_{a})-\mu\otimes{\mathcal{K}}(a,\ell_{a})\Big{|}>\varepsilon\Big{\\}}=0$

and

$\lim_{\lambda\to\infty}\mathbb{P}\Big{\\{}\sup_{([y_{u},\ell_{u}],[y_{v},\ell_{v}])\in{\mathcal{W}}\times{\mathcal{W}}}\Big{|}M_{2}^{Y^{\lambda}}([u,\ell_{u}],[y_{v},\ell_{v}])-t\mu\otimes{\mathcal{K}}\times\mu\otimes{\mathcal{K}}([y_{u},\ell_{u}],[y_{v},\ell_{v}])\Big{|}>\varepsilon\Big{\\}}=0$

###### Proof.

Let

$U_{1,{\mathcal{W}}}=\Big{\\{}\beta:\sup_{(a,\ell_{a})\in{\mathcal{W}}}|\beta(a,\ell_{a})-\mu\otimes{\mathcal{K}}(a,\ell_{a})|>\varepsilon\Big{\\}},$
$U_{2,{\mathcal{W}}}=\Big{\\{}\phi:\sup_{([u,\ell_{u}],[j,\ell_{v}])\in{\mathcal{W}}\times{\mathcal{W}}}|\phi([y_{u},\ell_{u}],[y_{v},\ell_{v}])-t\mu\otimes{\mathcal{K}}\times\mu\otimes{\mathcal{K}}([y_{u},\ell_{u}],[y_{v},\ell_{v}])|>\varepsilon\Big{\\}}$

and $U_{3,{\mathcal{W}}}=U_{1,{\mathcal{W}}}\cup U_{2,{\mathcal{W}}}.$ Now,
observe from Theorem 2.1 that

$\lim_{\lambda\to\infty}\frac{1}{\lambda}\log\mathbb{P}\Big{\\{}(M_{1}^{Y^{\lambda}},M_{2}^{Y^{\lambda}})\in
U_{3,{\mathcal{W}}}^{c}\Big{\\}}\leq-\inf_{(\beta,\phi)\in
F_{3,{\mathcal{W}}}^{c}}I(\beta,\phi).$

It meets the requirement for the study to prove that $I$ is strictly positive.
For instance,there is a sequence $(\beta_{n},\phi_{n})\to(\beta,\phi)$ such
that $I(\beta_{\lambda},\phi_{\lambda})\downarrow I(\beta,\phi)=0.$ This means
$\beta=\mu\otimes{\mathcal{K}}$ and
$\phi=t\mu\otimes{\mathcal{K}}\times\mu\otimes{\mathcal{K}}$ which contradicts
$(\beta,\phi)\in U_{3}^{c}.$ This ends the proof of the Lemma. ∎

We write $M_{\Delta}^{\lambda}=\frac{1}{\lambda}\sum_{u\in
I}\delta_{(\sigma^{\lambda},\sigma^{\lambda})}$ and observe that the
distribution of the marked SINR random network
$P(y)=\mathbb{P}\Big{\\{}Y^{\lambda}=y\Big{\\}}$ is given by

$P_{\lambda}(u)=\prod_{u=1}^{I}|\mu\otimes{\mathcal{K}}(y_{u},\ell_{u})\prod_{(u,v)\in
E}\frac{T^{r^{\lambda}}([y_{u},\ell_{u}],[y_{v},\ell_{v}])}{1-T^{\lambda}([y_{u},\ell_{u}],[y_{v},\ell_{v}])}\prod_{(u,v)\in{\mathcal{E}}}(1-T^{\lambda}([y_{u},\ell_{u}],[y_{v},\ell_{v}]))\prod_{x=1}^{I}(1-T^{\lambda}([y_{u},\ell_{u}],[y_{v},\ell_{v}]))$

$\displaystyle-\frac{1}{a_{\lambda}\lambda^{2}\log\lambda}\log P_{\lambda}(y)$
$\displaystyle=\frac{1}{a_{\lambda}\lambda\log\lambda}\Big{\langle}-\log\mu\otimes
T\,,M_{1}^{Y^{\lambda}}\Big{\rangle}+\frac{1}{\log\lambda}\Big{\langle}-\log\Big{(}\mbox{$\frac{T^{r^{\lambda}}}{1-T^{r^{\lambda}}}$}\Big{)}\,,M_{2}^{Y^{\lambda}}\Big{\rangle}$
$\displaystyle+\frac{1}{a_{\lambda}\log\lambda}\Big{\langle}-\log(1-T^{r^{\lambda}})\,,M_{1}^{Y^{\lambda}}\otimes
M_{1}^{Y^{\lambda}}\Big{\rangle}+\frac{1}{a_{\lambda}\lambda\log\lambda}\Big{\langle}-\log(1-T^{\lambda})\,,M_{\Delta}^{\lambda}\Big{\rangle}$

Notice,

$\displaystyle\lim_{\lambda\to\infty}\frac{1}{a_{\lambda}\lambda\log\lambda}\Big{\langle}-\log\mu\otimes{\mathcal{K}}\,,M_{1}^{Y^{\lambda}}\Big{\rangle}=\lim_{\lambda\to\infty}\frac{1}{\lambda}\Big{\langle}-\log(1-T^{\lambda}\,,M_{\Delta}^{\lambda}\Big{\rangle}=\lim_{\lambda\to\infty}\frac{1}{a_{\lambda}\log\lambda}\Big{\langle}-\log(1-T^{r^{\lambda}})\,,M_{1}^{Y^{\lambda}}\otimes
M_{1}^{Y^{\lambda}}\Big{\rangle}=0.$

Using, Lemma 4.1 we have

$\lim_{\lambda\to\infty}\frac{1}{\log\lambda}\Big{\langle}-\log\Big{(}T^{\lambda}/(1-T^{\lambda}\Big{)}\,,M_{2}^{Y^{\lambda}}\Big{\rangle}=\Big{\langle}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}\,,t\mu\otimes{\mathcal{K}}\times\mu\otimes{\mathcal{K}}\Big{\rangle}$

which concludes the proof of Theorem 2.2.

##  5\. Proof of Theorem 2.3, Corollary 2.4, Corollary 2.5

For $\beta\in{\mathcal{M}}({\mathcal{W}})$ we define the spectral potential of
the marked SINR graph $(Y^{\lambda})$ conditional on the event
$\big{\\{}M_{1}^{Y^{\lambda}}=\beta\big{\\}},$ $\rho_{t}(q,\beta)$ as

$\rho_{t}(q,\beta)=\Big{\langle}-(1-e^{q})\,,\,t\beta\otimes\beta\Big{\rangle}.$
(5.1)

Note that remarkable properties of a spectral potential, see [1] or [9] holds
for $\rho_{t}$.

For $\beta\in{\mathcal{M}}({\mathcal{W}}\times{\mathcal{W}})$, we observe that
$I_{\beta}(\phi)$ is the Kullback action of the marked SINR graph
$Y^{\lambda}$.

######  Lemma 5.1.

The following hold for the Kullback action or divergence function
$I_{\beta}(\phi)$:

  * •

$I_{\beta}(\phi)=\sup_{g\in{\mathcal{C}}}\big{\\{}\langle
g,\,\phi\rangle-\phi_{t}(g,\beta)\big{\\}}$

  * •

The function $I_{\beta}(\phi)$ is convex and lower semi-continuous on the
space ${\mathcal{M}}({\mathcal{W}}\times{\mathcal{W}}).$

  * •

For any real $\alpha$, the set
$\Big{\\{}\phi\in{\mathcal{M}}({\mathcal{W}}\times{\mathcal{W}}):\,I_{\beta}(\phi)\leq\alpha\Big{\\}}$
is weakly compact.

The proof of Lemma 5.1 is excluded from the article. Scholars of interest may
infer to [10] for likewise proof for empirical measures of ‘ the supercritical
marked SINR random network processes and/or the references therein for proof
of the lemma for empirical measures on measurable spaces.

Note from Lemma 5.1 that, for any $\varepsilon>0$, there exists some function
$q\in{\mathcal{W}}\times{\mathcal{W}}$ such that

$I_{\beta}(\phi)-\mbox{$\frac{\varepsilon}{2}$}<\langle
q\,,\,\beta\rangle-\phi_{t}(q,\phi).$

We define the probability distribution of the powered $R$ by $P_{\beta}$ by

$P_{\beta}(y)=\prod_{(u,v)\in
E}e^{q(u,v)}\prod_{(u,v)\in{\mathcal{E}}}e^{g_{\lambda}(u,v)},$

where

$g_{\lambda}(u,v)=\frac{1}{a_{\lambda}}\log\Big{[}1-T^{\lambda}(u,v)+T^{\lambda}(u,v)e^{q(u,v)}\Big{]}$

Then, clearly that

$\displaystyle\frac{dP_{\beta}}{d\tilde{P}_{\beta}}(y)$
$\displaystyle=\prod_{(u,v)\in
E}e^{-q(u,v)}\prod_{(u,v)\in{\mathcal{E}}}e^{-g_{\lambda}(u,v)a_{\lambda}}$
$\displaystyle=e^{-\lambda^{2}a_{\lambda}(\langle\mbox{$\frac{1}{2}$}q,M_{2}^{Y^{\lambda}}\rangle-\lambda^{2}a_{\lambda}\langle\mbox{$\frac{1}{2}$}h_{\lambda},M_{1}^{Y^{\lambda}}\otimes
M_{1}^{Y^{\lambda}}\rangle)+\langle\mbox{$\frac{1}{2}$}g_{\lambda},M_{\Delta}^{\lambda}\rangle}$

Now define the neighbourhood of $\phi,$ $B_{\phi}$ by

$B_{\phi}:=\Big{\\{}\omega\in{\mathcal{M}}({\mathcal{W}}\times{\mathcal{W}}):\,\langle
q,\omega\rangle-\rho_{t}(q,\beta)>\langle
q,\phi\rangle-\rho_{t}(q,\phi)-\varepsilon/2\Big{\\}}$

Note that under the condition $M_{2}^{Y^{\lambda}}\in B_{\phi}$ we have

$\displaystyle\frac{dP_{\beta}}{d\tilde{P}_{\beta}}(y)<e^{-\lambda^{2}a_{\lambda}(\langle\mbox{$\frac{1}{2}$}g,L_{2}^{\lambda}\rangle-\lambda^{2}a_{\lambda}\langle\mbox{$\frac{1}{2}$}h_{\lambda},M_{1}^{Y^{\lambda}}\otimes
M_{1}^{Y^{\lambda}}\rangle)+\langle\mbox{$\frac{1}{2}$}h_{\lambda},M_{\Delta}^{\lambda}\rangle}<e^{-\lambda^{2}a_{\lambda}I_{sc}(\nu)+\lambda^{2}a_{\lambda}\varepsilon}$

Thus, the study can deduce that

$P_{\beta}\Big{\\{}Y^{\lambda}\in{\mathcal{G}}\Big{|}M_{2}^{Y^{\lambda}}\in
D_{\phi}\Big{\\}}\leq\int{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{M_{2}^{Y^{\lambda}}\in
D_{\phi}\\}}d\tilde{P}_{\beta}(Y^{\lambda})\leq\int
e^{-\lambda^{2}a_{\lambda}I_{sc}(\beta)-\lambda\varepsilon}d\tilde{P}_{\beta}(Y^{\lambda})\leq
e^{-\lambda^{2}a_{\lambda}I_{sc}(\phi)-\lambda^{2}a_{\lambda}\varepsilon}.$

.

Given that $I^{2}(\phi)=0$ means Theorem 2.2 (ii), hence it is enough us to
obtain that the result is true for a probability distribution of the form
$\phi=e^{q}\beta\otimes\beta$ and for
$I^{2}(\phi)=\mbox{$\frac{1}{2}$}{\mathcal{H}}(\phi\|t\beta\otimes\beta),$
where $\beta=\mu\otimes{\mathcal{K}}$. Fix any number $\varepsilon>0$ and any
neigbourhood $B_{\phi}\subset{\mathcal{M}}({\mathcal{W}}\times{\mathcal{W}})$.
Now define the sequence of sets

${\mathcal{G}}^{\lambda}=\Big{\\{}y^{\lambda}\in{\mathcal{G}}:M_{2}^{y^{\lambda}}\in
B_{\phi}\Big{|}\langle
q,M_{2}^{y^{\lambda}}\rangle-\rho_{t}(q,\beta)\Big{|}\leq\mbox{$\frac{\varepsilon}{2}$}\Big{\\}}.$

Note that for all $q\in{\mathcal{G}}^{\lambda}$ we have

$\begin{aligned}
\frac{dP_{\beta}}{d\tilde{P}_{\beta}}(y)>e^{-\lambda^{2}a_{\lambda}\langle\mbox{$\frac{1}{2}$}q,\phi\rangle+\lambda^{2}a_{\lambda}\phi_{t}(q,\,\beta)+\lambda^{2}a_{\lambda}\mbox{$\frac{\varepsilon}{2}$}}\end{aligned}.$

This yields

$P_{\pi}({\mathcal{G}}^{\lambda})=\int_{{\mathcal{G}}^{\lambda}}dP_{\beta}(y)\geq\int
e^{-\lambda^{2}a_{\lambda}\langle\mbox{$\frac{1}{2}$}g,\nu\rangle+\lambda^{2}a_{\lambda}\rho_{t}(g,\,\beta)+\lambda^{2}a_{\lambda}\mbox{$\frac{\varepsilon}{2}$}}d\tilde{P}_{\beta}(y)\geq
e^{-\lambda^{2}a_{\lambda}\mbox{$\frac{1}{2}$}{\mathcal{H}}(\nu\|t\beta\otimes\beta)+\lambda^{2}a_{\lambda}\varepsilon}\tilde{P}_{\beta}({\mathcal{G}}^{\lambda}).$

Applying the law of large numbers, we have that
$\lim_{\lambda\to\infty}\tilde{P}_{\beta}({\mathcal{G}}^{\lambda})=1.$ This
completes of the Theorem.

Proof of Corollary 2.4

The proof of Corollary 2.4 follows from the definition of the Kullback action
and Theorem 2.3 if we set $\beta=\mu\otimes{\mathcal{K}}$ and
$\lambda\beta\otimes\beta(a,b)=\|\lambda\beta\otimes\beta\|,$ for all
$(a,b)\in{\mathcal{Y}}\times{\mathcal{Y}}.$

Proof of Corollary 2.5

In this scenario, the result was obtained by Lemma 3.3 the law of empirical
link measure is exponentially tight. Moreover, without loss of generality, we
can assume that the set $U$ in Corollary 2.5(ii) above is relatively compact.
If the study chooses any $\varepsilon>0$; then for each functional $\phi\in U$
the researchers can find a weak neighborhood such that the estimate of Theorem
2.3(u) above holds. From all these neighborhood, the study select a finite
cover of ${\mathcal{G}}$ and sums up over the value in Corollary 2.5(u) above
to obtain

$\limsup_{\lambda\to\infty}\frac{1}{\lambda}\log\mathbb{P}_{\beta}\Big{\\{}Y^{\lambda}\in{\mathcal{G}}\,\Big{|}\,M_{2}^{\lambda}\in
U\Big{\\}}\leq-\inf_{\phi\in U}I_{\beta}(\phi)+\varepsilon,\,\,\mbox{ where
$\beta=\mu\otimes{\mathcal{K}}$.}$

As $\varepsilon$ was arbitrarily chosen and the lower bound in Theorem 2.1(ii)
means in the lower bound in Theorem 2.5 holds, the study obtains the desired
results which completes the proof.

##  6\. Conclusion

The study provided a joint large deviation principle for the empirical power
measure and the empirical connectivity measure of telecommunication networks
in the $\tau-$ topology. Adopting the concept of the large deviations,we have
proved Shannon-McMillian Breiman Theorem for the telecommunication network
modelled as the sub-critical SINR network model. In addition, we have proved a
local large deviation principle for the empirical connectivity measure given
the empirical power measure and from this result;we have obtained the
classical McMillian theorem and for a given PPP. Finally, we have obtained an
asymptotic bound on the set of all possible sub-critical SINR network
processes . Conclusively, we have presented large deviation principles for the
sub-critical SINR networks. Note, that our results may form the bases for
designing an anomaly inference algorithms for subcritical wireless
telecommunication network models.

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