Large Deviations, Sharron-McMillan-Breiman Theorem for Super-Critical Telecommunication Networks
aa r X i v : . [ m a t h . P R ] N ov LARGE DEVIATIONS, SHARRON-MCMILLAN-BREIMANTHEOREM FOR SUPER-CRITICAL TELECOMMUNICATIONNETWORKS
By E. Sakyi-Yeboah , P. S. Andam , L. Asiedu and K. Doku-Amponsah , Department of Statistics and Actuarial Science, University of Ghana, BOX LG 115, Legon,Accra Email: [email protected] Telephone: +233205164254
Abstract.
In this article we obtain large deviation asymptotics for supercriticalcommunication networks modelled as signal-interference-noise ratio networks. To dothis, we define the empirical power measure and the empirical connectivity measure,and prove joint large deviation principles(LDPs) for the two empirical measures ontwo different scales i.e. λ and λ a λ , where λ is the intensity measure of the poissonpoint process (PPP) which defines the SINR random network.Using this joint LDPswe prove an asymptotic equipartition property for the stochastic telecommunicationNetworks modelled as the SINR networks. Further, we prove a Local large deviationprinciple(LLDP) for the SINR Network. From the LLDP we prove the a large devi-ation principle, and a classical McMillian Theorem for the stochastic SNIR networkprocesses. Note, for tupical empirical connectivity measure, qπ ⊗ π, we can deducefrom the LLDP a bound on the cardinality of the space of SINR networks to be ap-proximately equal to e λ a λ k qπ ⊗ π k H (cid:0) qπ ⊗ π/ k qπ ⊗ π k (cid:1) , where the connectivity probabilityof the network, Q z λ , satisfies a − λ Q z λ → q. Observe, the LDP for the empirical mea-sures of the stochastic SINR network were obtained on spaces of measures equippedwith the τ − topology, and the LLDPs were obtained in the space of SINR networkprocess without any topological restrictions. Keywords:
Super-critical sinr networks, Poisson Point Process, Empirical power measure, Empiricalconnectivity measure, Large deviations, Relative entropy, Entropy
AMS Subject Classification:
Introduction and Background . Large deviations may be regarded as a group of mathematical techniques(stochastic methods) often use to estimate asymptotic properties of increasingly rare events such astheir empirical measures and most likely manner of occurrence. See, for example, [13]. There aremany applications of large deviation techniques to SINR networks as a model for Telecommunicationnetworks. Some of this applications include but not limited to the analysis of bi-stability in networks,example notorious bi-stability in multiple access protocols such as the Aloha, and the stochastic
Acknowledgement:
This Research work has been supported by funds from the Carnegie Banga-Africa Project,University of Ghana
LLDP AND LDP FOR SUPER-CRITICAL COMMUNICATION NETWORKS behaviour of ATM such as the admission control, sizing of internal buffers, and the simulation ofATM models. See,[13].The Shanno-MacMillian-Breiman (SMB) Theorem or tte asymptotic equipartition property may beregarded as the strong law of large numbers in information theory. It says output source of a stochasticdata source may be partition into two sets, namely the set of typical events and the set of atypicalevents. The SMB is the foundation of all approximate pattern matching and coding algorithms.Researchers over the last two decades have given some large deviation analysis for telecommnunicationnetworks modelled as a sequence of i.i.d random variables and or markov chains in discrete andcontinuous times. See, [13] and reference therein. [11] and [12] defined empirical measures on theSINR network and proved some jonit LDP results including the SMB and the classical MacMilliantheorem for the dense or critical telecommunication networks modelled as the SINR network.In this article we prove joint large deviation principles on the scales λ and λ a λ , where λ is theintensity measure of the underlining PPP of the SINR network. See, [2] or [3] for similar results forethe colored random graph models.From these LDPs we prove an asymptotic equipartition property,see example [8], for the SINR networks.Further, we prove a local LDP for the SINR networks. See example, [10] or [9] and reference therein.From the local LDP we deduce asymptotic bounds on the cardinality of the set of SINR networks fora given typical empirical power measure. We also prove from the local LDP an LDP for the SINRnetwork processes.The remaining part of the article is organized in this manner: Section 2 contains the main results;Theorem 2.1, Theorem 2.2 Theorem 2.3, Theorem 2.4, Corollary 2.5 and Corollary 2.6. In Section 3the main results of the article, Theorem 2.1. Section 3.3 contain proof of the SBM, see Theorem 2.3and Section 5; Proof of Theorem 2.4, Corollary 2.5 and Corollary 2.6. Finally. we give the conclusionof the article in Section 6 We fix dimension d ∈ N and some measureable set D ⊂ R d with respect to the Borel-Sgma algebra B ( R d ) . For an intensity function, λπ : D → [0 , D to (0 , ∞ ) , K and apath loss function, β ( ℓ ) = ℓ − r , where r ∈ (0 , ∞ ) , and some technical constants; τ ( λ ) , γ ( λ ) : (0 , ∞ ) → (0 , ∞ ) , we define the Sinr network model as follows: • We pick Z = ( Z i ) i ∈ I a Poisson Point process (PPP) with rate measure λπ : D → [0 , • Given Z, we assign each Z i a power η ( Z i ) = η i independently according to the transitionfunction K ( · , Z i ) . • For any two powered points (( Z i , η i ) , ( Z j , η j )) we connect a link iff SIN R ( Z i , Z j , Z ) ≥ τ ( λ ) ( η j ) and SIN R ( Z j , Z i , Z ) ≥ τ ( λ ) ( η i ) , where SIN R ( Z j , Z i , Z ) = η i β ( k Z i − Z j k ) N + γ ( λ ) ( η j ) P i ∈ I \{ j } η i β ( k Z i − Z j k )We shall consider Z λ := Z λ ( η, K , β ) = n [( Z i , η i ) , j ∈ I ] , E o under the joint law of the powered PoissonPoint Process and the Network. We will interpret Z λ as an SINR Network and ( Z i , η i ) := Z λi as the LDP AND LDP FOR SUPER-CRITICAL COMMUNICATION NETWORKS 3 power type of device i. We recall from [11] that the link/connectivity probability of the SINR network, Q z λ , is given by Q z λ (( x, η x ) , ( y, η y )) = e − λq D λ (( x,η x ) , ( y,η y ))) , where q D λ (( x, η x ) , ( y, η y )) = Z D h τ ( λ ) ( η x ) γ ( λ ) ( η x ) τ ( λ ) ( η x ) γ ( λ ) ( η x )+( k z k ℓ / k x − y k ℓ ) + τ ( λ ) ( η y ) γ ( λ ) ( η y ) τ ( λ ) ( η y ) γ ( λ ) ( η y )+( k z k ℓ / k y − x k ℓ ) i π ( dz ) . We have assumed there exists a sequence of real numbers, a λ and a function q : D × R + → (0 , ∞ ) suchthat λ a λ → ∞ and lim λ ↑∞ a − λ Q z λ (( x, η x ) , ( y, η y )) = q (( x, η x ) , ( y, η y )) . Sakyi-Yeboah et. al [12] studied the critical SINR Networks (i.e. λa λ → λ →∞ λa λ → ∞ ).We define the set S ( D ) by S ( D ) = ∪ x ⊂D n x : | x ∩ A | < ∞ , for any bounded A ⊂ D o . (1.1)Write W = S ( D × R + ) and M ( W ), denote the space of positive measures on the space W equippedwith τ − topology. Note, W a locally finite subset of the set W . See, example, [12] or [4] For any SINRNetwork Z λ we define a probability measure, the empirical power measure , U λ ∈ M ( W ), by U λ (cid:0) ( x, η x ) (cid:1) := 1 λ X i ∈ I δ Z λi (cid:0) ( x, η x ) (cid:1) and a finite measure, the empirical connectivity measure U λ ∈ M ( W × W ) , by U λ (cid:0) ( x, η x ) , ( y, η y ) (cid:1) := 1 λ a λ X ( i,j ) ∈ E [ δ ( Z λi ,Z λj ) + δ ( Z λj ,Z λi ) ] (cid:0) ( x, η x ) , ( y, η y ) (cid:1) . Note that the total mass k U λ k of the empirical power measure is 1l and total mass of the empiricallink measure is 2 | E | /λ a λ . 2. Main Results
Theorem 2.1, is a Joint Large deviation principle for the empirical measures of the Sinr networkmodels.We recall from Subsection 1.2 the definition of q D λ as q D λ (( x, η x ) , ( y, η y )) = Z D h τ ( λ ) ( η x ) γ ( λ ) ( η x ) τ ( η x ) γ ( η x )+( k z k ℓ / k x − y k ℓ ) + τ ( λ ) ( η y ) γ ( λ ) ( η y ) τ ( λ ) ( η y ) γ ( λ ) ( η y )+( k z k ℓ / k y − x k ℓ ) i η ( dz )and write qπ ⊗ π (( x, η x ) , ( y, η y ))) := q (( x, η x ) , ( y, η y )) µ (( x, η x )) µ (( y, η y )) . Theorem 2.1.
Let Z λ is a super critical powered Sinr network with rate measure λη : D → [0 , anda power probability function K ( · , η ) = ce − cη , η ≥ and path loss function β ( r ) = r − ℓ , for ℓ > . Thus,the connectivity probability Q z λ of Z λ satisfies a − λ Q z λ → q and λa λ → ∞ . Then, as λ → ∞ , the pairof measures ( U λ , U λ ) satisfies a large deviation principle in the space M ( W ) × M ( W × W )(i) with speed λ and good rate function I Sc (cid:0) π, ν (cid:1) = ( H (cid:16) π (cid:12)(cid:12)(cid:12) η ⊗ q (cid:17) if ν = qπ ⊗ π ∞ otherwise. (2.1) LLDP AND LDP FOR SUPER-CRITICAL COMMUNICATION NETWORKS (ii) with speed λ a λ and good rate function I Sc (cid:0) π, ν (cid:1) = 12 H (cid:16) ν k qπ ⊗ π (cid:17) (2.2) where H ( ν k qπ ⊗ π ) := ( H ( ν k qπ ⊗ π ) + (cid:16) k qπ ⊗ π k − k ν k (cid:17) , if k ν k > . ∞ otherwise. (2.3) and qπ ⊗ π (( x, η x ) , ( y, η y ))) = q (( x, η x ) , ( y, η y )) π (( x, η x )) π (( y, η y )) . Theorem 2.2 below is a key step in the proof of Theorem 2.1. See subsection 3.3.
Theorem 2.2.
Let Z λ is a super critical powered Sinr network with rate measure λη : D → [0 , anda power probability function K ( · , η ) = ce − cη , η ≥ and path loss function β ( r ) = r − ℓ , for ℓ > . Thus,the connectivity probability Q z λ of Z λ satisfies a − λ Q z λ → q and λa λ → ∞ . Let Z λ be a super criticalpowered Sinr network conditional on event (cid:8) U λ = π, (cid:9) . Then, as λ → ∞ , the pair of measures U λ satisfies a large deviation principle in the space M ( W ) × M ( W × W )(i) with speed λ and good rate function I π (cid:0) ν (cid:1) = (cid:26) if ν = qπ ⊗ π ∞ otherwise. (2.4)(ii) with speed λ a λ and good rate function I π (cid:0) ν (cid:1) = 12 H (cid:16) ν k qπ ⊗ π (cid:17) . (2.5) Theorem 2.3.
Let Z λ is a super critical powered Sinr network with rate measure λµ : D → [0 , anda power probability function K ( η ) = ce − cη , η > and path loss function β ( r ) = r − ℓ , for ℓ > . Thus,the connectivity probability Q z λ of Z λ satisfies a − λ Q z λ → q and λa λ → ∞ . Suppose the sequence a λ of Z λ is such that λa λ log λ → ∞ and a λ / log λ → − . Then, we have lim λ →∞ P n(cid:12)(cid:12)(cid:12) − a λ λ log λ log P ( Z λ ) − Z W×W q (( x, η x ) , ( y, η y )) q ( dη x ) q ( dη y ) dxdy (cid:12)(cid:12)(cid:12) ≥ ε o = 0 . Theorem 2.4.
Let Z λ is a super critical powered Sinr network with rate measure λµ : D → [0 , anda power probability function K ( η ) = ce − cη , η > and path loss function β ( r ) = r − ℓ , for ℓ > . Thus,the connectivity probability Q z λ of Z λ satisfies a − λ Q z λ → q and λa λ → ∞ . Then, • for any functional ν ∈ M π and a number ε > , there exists a weak neighbourhood B ν suchthat P π n Z λ ∈ G P (cid:12)(cid:12)(cid:12) L λ ∈ B ν o ≤ e − λ a λ H ( ν k qπ ⊗ π ) − λa λ ε . • for any ν ∈ M π , a number ε > o and a fine neighbourhood B ν , we have the estimate: P π n Z λ ∈ G P (cid:12)(cid:12)(cid:12) L λ ∈ B ν o ≥ e − λ a λ H ( ν k qπ ⊗ π )+ λ λ a λ ε . We define for telecommunication networks an entropy h : M ( W × W ) → [0 . ∞ ] by h ( ν ) := (cid:16) k ν k − k λπ ⊗ π k − D ν , log ν k qπ ⊗ π k E(cid:17) / . (2.6) LDP AND LDP FOR SUPER-CRITICAL COMMUNICATION NETWORKS 5
Corollary 2.5 (McMillian Theorem) . Let G p be a super critical powered Sinr network with ratemeasure λµ : D → [0 , and a power probability function K ( η ) = ce − cη , η > and path loss function β ( r ) = r − ℓ , for ℓ > . Thus, the connectivity probability Q z λ of every z λ ∈ G p satisfies a − λ Q z λ → q and λa λ → ∞ . (i) For any empirical connectivity measure ν on W × W and ε > , there exists a neighborhood B ν such that Card (cid:16)(cid:8) z λ ∈ G p | L λ ∈ B ν (cid:9)(cid:17) ≥ e λ a λ ( h ( ν ) − ε (cid:1) . (ii) for any neighborhood B ρ and ε > , we have Card (cid:16)(cid:8) z λ ∈ G p | U λ ∈ B ν (cid:9)(cid:17) ≤ e λ a λ ( h ( ν )+ ε (cid:1) , where Card ( A ) means the cardinality of A. remark ν = qπ ⊗ π, we have Card (cid:16)n y ∈ G p o(cid:17) ≈ e λ a λ k qπ ⊗ π k h (cid:0) qπ ⊗ π/ k qπ ⊗ π k (cid:1) . Corollary 2.6.
Let Z λ is a super critical powered Sinr network with rate measure λη : D → [0 , anda power probability function K ( η ) = ce − cη , η > and path loss function β ( r ) = r − ℓ , for ℓ > . Thus,the connectivity probability Q z λ of Z λ satisfies a − λ Q z λ → q and λa λ → ∞ . • Let F be closed subset M π . Then we have lim sup λ →∞ λ a λ log P π n Z λ ∈ G P (cid:12)(cid:12)(cid:12) U λ ∈ F o ≤ − inf π ∈ F n H ( ν k qπ ⊗ π ) o . • Let O be open subset M p . Then we have lim inf λ →∞ λ a λ log P π n Z λ ∈ G p (cid:12)(cid:12)(cid:12) U λ ∈ O o ≥ − inf ν ∈ O n H ( ν k qπ ⊗ π ) o . Proof of Theorem 2.1 by Gartner-Ellis Theorem and Method of Mixtures
Suppose A , ..., A n is a decomposition of the space D × R + . Observe that, for every ( x, y ) ∈ A i × A j , i, j = 1 , , , ..., n, λU λ ( x, y ) given λU λ ( x ) = λµ ( x ) is binomial with parameters λ µ ( x ) µ ( y ) / Q z λ ( x, y ) . Let q be the exponential distribution with parameter c. We recall the function q D λ from theprevious sections and note that Lemma 2.4 is key component in the application of the Gartner-EllisTheorem. See [1]. Lemma 3.1.
Let Z λ is a super critical powered Sinr network with rate measure λµ : D → [0 , anda power probability function K ( η ) = ce − cη , η > and path loss function β ( r ) = r − ℓ , for ℓ > . Thus,the connectivity probability Q z λ of Z λ satisfies a − λ Q z λ → q and λa λ → ∞ . Let Z λ be a supercriticalSINR network, conditional on the event U λ = π. Let g : W × W → R be bounded function. Then, lim λ →∞ λ log E n e λ h g, U λ i (cid:12)(cid:12)(cid:12) U λ = π o = 12 lim n →∞ n X j =1 n X i =1 D g, qπ ⊗ π E A i × A j = 12 D g, qπ ⊗ π E W×W . LLDP AND LDP FOR SUPER-CRITICAL COMMUNICATION NETWORKS
Proof.
Now we observe that E n e R R λg ( x,y ) U λ ( dx,dy ) / (cid:12)(cid:12)(cid:12) U λ = π o = E n Y x ∈W Y y ∈W e λg ( x,y ) U λ ( dx,dy ) / o E n Y x ∈W Y y ∈W e g ( x,y ) λU λ ( dx,dy/ o = Y i =1 Y j =1 Y x ∈ A i Y y ∈ A j E n e g ( x,y ) λU λ ( dx,dy ) / o log n e λ h g,U λ i / (cid:12)(cid:12)(cid:12) U λ = π o = n X j =1 n X i =1 Z B j Z B i log h − Q z λ ( x, y ) + Q z λ ( x, y ) e g ( x,y ) /λa λ i λ π ⊗ π ( dx,dy ) / + o ( n )By the dominated convergence theorem1 λ log E { e λ h g,U λ i / | U λ = π } = 1 λ X j =1 X i =1 Z A i Z A j log h − (cid:0) − e g ( x,y ) /λa λ ) Q z λ ( x, y ) i λ π ⊗ π ( dx,dy ) / + o ( n ) /λ λ log E { e λ h g,U λ i / (cid:12)(cid:12) U λ = π } = lim λ →∞ X j =1 X i =1 Z A i Z A j log h g ( x, y ) q ( x, y ) /λ + o ( λ ) /λ i λπ ⊗ π ( dx,dy ) / + o ( n ) /λ lim λ →∞ λ log E { e λ h g,U λ i / | U λ = π } = 12 n X j =1 n X i =1 D g, qπ ⊗ π E A i × A j lim λ →∞ λ log E { e λ h g,U λ i / (cid:12)(cid:12)(cid:12) U λ = π } = 12 lim n →∞ n X j =1 n X i =1 D g, qπ ⊗ π E A i × A j = 12 D g, qπ ⊗ π E W×W . Hence, by Gartner-Ellis theorem, conditional on the event n U λ = µ o , U λ obey a large deviationprinciple with speed λ and variational formulation of the rate function I µ ( π ) = 12 sup g nD g, π E W×W − D g, qπ ⊗ π E W×W o which when solved, see example [2], would clearly reduces to the good rate function given by I π ( ν ) = 0 . Similarly we take A , ..., A n a a decomposition of the space D × R + . We recall the function h Dλ fromthe previous sections and state the following Lemma. Lemma 3.2 is key component in the applicationof the Gartner-Ellis Theorem. See, [1]. Lemma 3.2.
Let Z λ is a super critical powered Sinr network with rate measure λµ : D → [0 , anda power probability function K ( η ) = ce − cη , η > and path loss function β ( r ) = r − ℓ , for ℓ > . Thus,the connectivity probability Q z λ of Z λ satisfies a − λ Q z λ → q and λa λ → ∞ . Let Z λ be a supercriticalSINR network, conditional on the event U λ = π. Let g : W × W → R be bounded function. Then, LDP AND LDP FOR SUPER-CRITICAL COMMUNICATION NETWORKS 7 lim λ →∞ λ a λ log E n e λ a λ h g, U λ i (cid:12)(cid:12)(cid:12) U λ = π o = −
12 lim n →∞ n X j =1 n X i =1 D − e g , qπ ⊗ π E A i × A j = − D − e g , qπ ⊗ π E W×W . Proof.
Now we observe that E n e R R λ a λ g ( x,y ) U λ ( dx,dy ) / (cid:12)(cid:12)(cid:12) U λ = π o = E n Y x ∈W Y y ∈W e λ a λ g ( x,y ) U λ ( dx,dy ) / o E n Y x ∈W Y y ∈W e g ( x,y ) λU λ ( dx,dy/ = Y i =1 Y j =1 Y x ∈ A i Y y ∈ A j E n e λ a λ g ( x,y ) U λ ( dx,dy ) / o × e o ( n ) log n e λ a λ h g,U λ i / (cid:12)(cid:12)(cid:12) U λ = π o = n X j =1 n X i =1 Z A j Z A i log h − Q z λ ( x, y )) + Q z λ ( x, y ) e g ( x,y ) i λ π ⊗ π ( dx,dy ) / + o ( n )By the dominated convergence theorem1 λ a λ log E { e λ h g,U λ i / | U λ = π } = 1 λ a λ X j =1 X i =1 Z A i Z A j log h − (cid:0) − e g ( x,y ) ) Q z λ ( x, y ) i λ π ⊗ π ( dx,dy ) / + o ( n ) /λ a λ λ a λ log E { e λ h g,U λ i / k U λ = π } = lim λ →∞ X j =1 X i =1 Z A i Z A j log h − (1 − e g ( x,y ) ) Q z λ ( x, y ) i λπ ⊗ π ( dx,dy ) / + o ( n ) /λ a λ lim λ →∞ λ a λ log E n e λ h g,U λ i / (cid:12)(cid:12) U λ = π o = − X j =1 X i =1 Z A i Z A j h (1 − e g ( x,y ) ) q ( x, y ) π ⊗ π ( dx, dy ) i lim λ →∞ λ a λ log E { e λ h g,U λ i / | U λ = π } = − n X j =1 n X i =1 D − e g , qπ ⊗ π E A i × A j lim λ →∞ λ a λ log E { e λ h g,U λ i / (cid:12)(cid:12)(cid:12) U λ = π } = −
12 lim n →∞ n X j =1 n X i =1 D − e g , qπ ⊗ π E A i × A j = − D − e g , qπ ⊗ π E W×W
Hence, by Gartner-Ellis theorem, conditional on the event n M λ = µ o , U λ obey a large deviationprinciple with speed λ and variational formulation of the rate function I µ ( π ) = 12 sup g nD g, π E W×W + D − e g , qπ ⊗ π E W×W o which when solved, see example [2], would clearly reduces to the good rate function given by I π ( ν ) = 12 H ( ν k qπ ⊗ π ) . (3.1) LLDP AND LDP FOR SUPER-CRITICAL COMMUNICATION NETWORKS
For any λ ∈ (0 , ∞ ) we define M λ ( W ) := n µ ∈ M ( W ) : λµ ( x ) ∈ N for all x ∈ W o , ˜ M λ ( W × W ) := n π ∈ ˜ M ( W × W ) : λ π ( x, y ) ∈ N , for all x, y ∈ W o . We denote by Θ λ := M λ ( W ) and Θ := M ( W ). We write P ( λ ) µ λ ( η λ ) := P (cid:8) U λ = η λ (cid:12)(cid:12) U λ = π λ (cid:9) ,P ( λ ) ( µ λ ) := P (cid:8) U λ = π λ (cid:9) Th joint distribution of U λ and U λ is the mixture of P ( λ ) µ λ with P ( λ ) ( µ λ ) , as follows: d ˜ P λ ( µ λ , η λ ) := dP ( λ ) µ n ( η λ ) dP ( λ ) ( µ λ ) . (3.2)(Biggins, Theorem 5(b), 2004) gives criteria for the validity of large deviation principles for the mix-tures and for the goodness of the rate function if individual large deviation principles are known. Thefollowing three lemmas ensure validity of these conditions.Observe that the family of measures ( P ( λ ) : λ ∈ (0 , ∞ )) is exponentially tight on Θ . Lemma 3.3. (i)
The family of measures ( ˜ P λ : λ ∈ (0 , ∞ )) is exponentially tight on Θ × ˜ M ∗ ( W ×W ) . (ii) The family measures ( Q z λ : λ ∈ (0 , ∞ )) is exponentially tight on Θ × ˜ M ∗ ( W × W ) . We refer to [11, Lemma 4.3] for similar proof for Large deviation Principle on the scale λ Define the function I Sc , I Sc : Θ × M ∗ ( W × W ) → [0 , ∞ ] , by I Sc (cid:0) π, ν (cid:1) = ( H (cid:16) π (cid:12)(cid:12)(cid:12) η ⊗ q (cid:17) if ν = qπ ⊗ π ∞ otherwise. (3.3) I Sc (cid:0) π, ν (cid:1) = 12 H (cid:16) ν k qπ ⊗ π (cid:17) . (3.4) Lemma 3.4. (i) I Sc is lower semi-continuous. (ii) I Sc is lower semi-continuous. By (Biggins, Theorem 5(b), 2004) the two previous lemmas, the LDP for the empirical power measure,see, [11, Theorem 2.1] and the large deviation principles we have established Theorem 2.2 ensure thatunder ( ˜ P λ ) and Q z λ the random variables ( π λ , η λ ) satisfy a large deviation principle on M ( W ) × ˜ M ( W × W ) and Θ × ˜ M λ ( W × W ) on the speeds λ and λ a λ with good rate functions I Sc and I Sc respectively, which ends the proof of Theorem 2.1.4. Proof of Theorem 2.3 by Large deviations
In order to establish the asymptotic equipartition property, we first prove a weak law of large numbersfor the empirical powere measure and the empirical connectivity measure of the SINR network.
LDP AND LDP FOR SUPER-CRITICAL COMMUNICATION NETWORKS 9
Lemma 4.1.
Let Z λ is a super critical powered Sinr network with rate measure λµ : D → [0 , anda power probability function K ( η ) = ce − cη , η > and path loss function β ( r ) = r − ℓ , for ℓ > . Thus,the connectivity probability Q z λ of Z λ satisfies a − λ Q z λ → q and λa λ → ∞ . Then, lim λ →∞ P n sup ( x,η x ) ∈W (cid:12)(cid:12)(cid:12) L λ ( x, η x ) − µ ⊗ K ( x, η x ) (cid:12)(cid:12)(cid:12) > ε o = 0 and lim λ →∞ P n sup ([ x,η x ] , [ y,η y ]) ∈W×W (cid:12)(cid:12)(cid:12) L λ ([ x, η x ] , [ y, η y ]) − qµ ⊗ K × µ ⊗ K ([ x, µ x ] , [ y, µ y ]) (cid:12)(cid:12)(cid:12) > ε o = 0 Proof.
Let F , W = n π : sup ( x,η x ) ∈W | π ( x, η x ) − m ⊗ K ( x, η x ) | > ε o ,F , W = n ν : sup ([ x,η x ] , [ y,η y ]) ∈W×W | ν ([ x, η x ] , [ y, η y ]) − qµ ⊗ K × µ ⊗ K ([ x, η x ] , [ y, η y ]) | > ε o and F , W = F , W ∪ F , W . Now, observe from Theorem 2.1 thatlim λ →∞ λ log P n ( L λ , L λ ) ∈ F c , W o ≤ − inf ( π,̟ ) ∈ F c , W I ( π, ̟ ) . It suffices for us to show that I is strictly positive. Suppose there is a sequence ( π n , ̟ n ) → ( π, ̟ )such that I ( π λ , ̟ λ ) ↓ I ( π, ̟ ) = 0 . This implies π = µ ⊗ K and ̟ = qµ ⊗ K × µ ⊗ K which contradicts( π, ̟ ) ∈ F c . This ends the proof of the Lemma. (cid:3)
Now, the distribution of the marked PPP P ( x ) = P n X λ = x o is given by P λ ( x ) = I Y i =1 | µ ⊗K ( x i , η i ) Y ( i,j ) ∈ E Q z λ ([ x i , µ i ] , [ y j , µ j ])1 − Q z λ ([ x i , µ i ] , [ y j , µ j ]) Y ( i,j ) ∈E (1 − Q z λ ([ x i , µ i ] , [ y j , µ j ])) I Y i =1 (1 − Q z λ ([ x i , µ i ] , [ y j , µ j ])) − a λ λ log λ log P λ ( x ) = 1 a λ λ log λ D − log µ ⊗ Q , L λ E + 1log λ D − log (cid:16) Q zλ − Q zλ (cid:17) , L λ E + 1 a λ log λ D − log(1 − Q z λ ) , L λ ⊗ L λ E + 1 a λ λ log λ D − log(1 − Q z λ ) , L λ ∆ E Notice,lim λ →∞ a λ λ log λ D − log µ ⊗K , L λ E = lim λ →∞ λ D − log(1 − Q z λ , L λ ∆ E = lim λ →∞ a λ log λ D − log(1 − Q z λ ) , L λ ⊗ L λ E = 0 . Using, Lemma 4.1 we havelim λ →∞ λ D − log (cid:16) Q z λ / (1 − Q z λ (cid:17) , L λ E = D , qµ ⊗ K × µ ⊗ K E which concludes the proof of Theorem 2.3. Proof of Theorem 2.4, Corollary 2.5, Corollary 2.6
For π ∈ M ( W ) we define the spectral potential of the marked SINR graph ( Z λ ) conditional on theevent (cid:8) L λ = π (cid:9) , ρ q ( g, π ) as φ q ( g, π ) = D − (1 − e g ) , qπ ⊗ π E . (5.1)Note that remarkable properties of a spectral potential, see or [11] holds for φ q .For π ∈ M ( W × W ), we observe that I π ( π ) is the Kullback action of the marked SINR graph Z λ . Lemma 5.1.
The following hold for the Kullback action or divergence function I π ( π ) : • I Sc ( π ) = sup g ∈C (cid:8) h g, π i − φ q ( g, π ) (cid:9) • The function I Sc ( π ) is convex and lower semi-continuous on the space M ( W × W ) . • For any real α , the set n π ∈ M ( W × W ) : I Sc ( π ) ≤ α o is weakly compact. The proof of Lemma 5.1 is omitted from the article. Interested readers may refer to [10] for similarproof for empirical measures of ‘ the Typed Random Graph Processes or See, for example [7] forthe multitype Galton-Watson processes and/or the reference therein, [5], for proof of the lemma forempirical measures on measurable spaces.Note from Lemma 5.1 that, for any ε >
0, there exists some function g ∈ W × W such that I Sc ( π ) − ε < h g , π i − φ q ( g, π ) . We define the probability distribution of the powered Z by P π by P π ( z ) = Y ( i,j ) ∈ E e g ( x i ,x j ) Y ( i,j ) ∈E e h λ ( x i ,x j ) , where h λ ( x, y ) = 1 a λ log h − Q z λ ( x, y ) + Q z λ ( x, y ) e g ( x,y ) i Then, observe that dP π d ˜ P π ( z ) = Y ( i,j ) ∈ E e − g ( x i ,x j ) Y ( i,j ) ∈E e − h λ ( x i ,x j ) a λ = e − λ a λ ( h g,L λ i− λ a λ h h λ ,L λ ⊗ L λ i )+ h h λ ,L λ ∆ i Now define the neighbourhood of ν, B ν by B ν := n ω ∈ M ( W × W ) : h g, ω i − ρ q ( g, π ) > h g, ν i − ρ q ( g, π ) − ε/ o Note that under the condition L λ ∈ B ν we have dP π d ˜ P π ( z ) < e − λ a λ ( h g,L λ i− λ a λ h h λ ,L λ ⊗ L λ i )+ h h λ ,L λ ∆ i < e − λ a λ I Sc ( ν )+ λ a λ ε LDP AND LDP FOR SUPER-CRITICAL COMMUNICATION NETWORKS 11
Therefore, we obtain P π n Z λ ∈ G P (cid:12)(cid:12)(cid:12) L λ ∈ B ν o ≤ Z { L λ ∈ B ν } d ˜ P π ( z λ )( z ) ≤ Z e − λ a λ I Sc ( π ) − λε d ˜ P π ( z λ ) ≤ e − λ a λ I Sc ( ν ) − λ a λ ε . .Note that I Sc ( ν ) = ∞ implies Theorem 2.3 (ii), hence it sufficient for us to deduce that the result istrue for a probability distribution of the form ν = e g π ⊗ π and for I Sc ( ν ) = H ( ν k qπ ⊗ π ) . Fix anynumber ε > B ν ⊂ M ( W × W ). Now define the sequence of sets G λp = n y ∈ G p : L λ ( y ) ∈ B ν (cid:12)(cid:12)(cid:12) h g, L λ i − φ q ( g, π ) (cid:12)(cid:12)(cid:12) ≤ ε o . Note that for all y ∈ G λp we have dP π d ˜ P π > e − λ a λ h g,ν i + λ a λ φ q ( g, π )+ λ a λ ε . This yields P π ( G λP ) = Z G λP dP π ( y ) ≥ Z e − λ a λ h g,ν i + λ a λ φ q ( g, π )+ λ a λ ε d ˜ P π ( y ) ≥ e − λ a λ H ( ν k qπ ⊗ π )+ λ a λ ε ˜ P π ( G λP ) . Applying the law of large numbers, we have that lim λ →∞ ˜ P π ( G λP ) = 1 . This completes of the Theorem.
Proof of Corollary 2.5
The proof of Corollary 2.5 follows from the definition of the Kullback action and Theorem 2.4 if weset π = ρ and λπ ⊗ π ( a, b ) = k λπ ⊗ π k , for all ( a, b ) ∈ Y × Y . Proof of Corollary 2.6
We observe that, by Lemma 3.3 the law of empirical connectivity measure is exponentially tight.Henceforth, without loss of generality we can assume that the set F in Corollary 2.6(ii) above isrelatively compact. If we choose any ε >
0; then for each functional ν ∈ F we can find a weakneigbourhood such that the estimate of Theorem 2.4(i) above holds. From all these neigbhoourhood,we choose a finite cover of G P and sum up over the estimate in Corollary 2.6(i) above to obtainlim sup λ →∞ λ log P π n Z λ ∈ G P (cid:12)(cid:12)(cid:12) L λ ∈ F o ≤ − inf ν ∈ F I π ( ν ) + ε. As ε was arbitrarily chosen and the lower bound in Theorem 2.6(ii) implies the lower bound in Theorem2.2(i) we get the desired results which completes the proof.6. Conclusion
In this article we have presented a joint large deviation principle for the empirical power measureand the empirical connectivity measure of telecommunication networks in the τ − topology. From thislarge deviation principle we deduce an asymptotic equipartition property for the telecommunicationnetwork modelled as the SINR network model.We have also presented a Local large deviation principle for the empirical connectivity measure giventhe empirical power measure and from this result we had deduce the classical MacMillian theorm andan asymptotic bound for the set alll posible SINR network process. Finaly, we the authors had also presented a large deviation principle for the SINR networks. This article might may be regarded asa first step in the proof of a Lossy asymptotic equipartition property for the SINR networks. See, [6]and [7] for similar results for the networked data structures modelled as colored random graph processand for the hierarchical data structure modelled as Galton-Watson tree process. References [1] Dembo, A. and Zetouni, O.(1998). Large Deviations Techniques and applicationsSpringers.[2] Doku-Amponsah, K.(2006). Large Deviations and Basic Information Theory for Hierar-chical and Networked Data Structures Ph.D Thesis, Bath.[3] Doku-Amponsah, K. and Moeters. P. (2010). Large deviation principle for empirical mea-sures of coloured random graphs.
Ann. Appl. Prob. 20(6),1089-2021. [4] Jahnel, B. and Konig,W. (2003). Probabilstic Methods in Telecommunication.
LectureNotes.
TU Berlin and WiAS Berlin.[5]
Bakhtin I.V..
Spectral Potential, Kullback Action, and Large deviations of empiricalmeasureson measureable spaces.
Theory of Probability and application. Vol. 50,No.4.(2015)pp.535-544. [6]
Doku-Amponsah, K.
Lossy Asymptotic Equipartition property for Networked DataStructures.
Journal of Mathematics and Statistics(JMSS) 13(2), pp.152-158 [7]
Doku-Amponsah, K.
Lossy Asymptotic Equipartition Property for hierarchical datastructures.
Far East Journal of Mathematical Sciences, Vol. 101, 2017, pp.1013-1024. [8]
Doku-Amponsah, K.
Asymptotic equipartition properties for hierarchical and networkedstructures. ESAIM: PS 16 (2012): 114-138.DOI: 10.1051/ps/2010016.[9]
Doku-Amponsah, K.
Local Large Deviations, McMillian Theorem for multitype Galton-Watson Processes .
Far East Journal of Mathematical Sciences, 2017, 102(10), pp. 2307-2319 .[10] Doku-Amponsah, K. (2017). Local Large deviation: A McMillian Theorem for ColouredRandom Graph Processes
Journal of Mathematics and Statistics 13(4) (2017) 347-352 .[11] Sakyi-Yeboah, E., Asiedu, L. and Doku-Amponsah, K.(2020) Local Large Deviation Prin-ciple, Large Deviation Principle and Information theory for the Signal -to- Interferenceand Noise Ratio Graph Models. To appear in
Journal of optimization and informationsciences [12] Sakyi-Yeboah, E.,Kwofie, C., Asiedu, L. and Doku-Amponsah, K.(2020) Large DeviationPrinciple for Empirical Sinr Measure of Critical Telecommunication Network. To appearin