Many-Help-One Problem for Gaussian Sources with a Tree Structure on their Correlation

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1 Many-Hep-One Probem for Gaussian Sources with a Tree Structure on their Correation Yasutada Oohama arxiv:090988v csit 4 Jan 009 Abstract In this paper we consider the separate coding probem for + correated Gaussian memoryess sources We dea with the case where separatey encoded data of sources work as side information at the decoder for the reconstruction of the remaining source The determination probem of the rate distortion region for this system is the so caed many-hep-one probem and has been known as a highy chaenging probem The author determined the rate distortion region in the case where the sources working as partia side information are conditionay independent if the remaining source we wish to reconstruct is given This condition on the correation is caed the CI condition In this paper we extend the author s previous resut to the case where + sources satisfy a kind of tree structure on their correation We ca this tree structure of information sources the TS condition which contains the CI condition as a specia case In this paper we derive an expicit outer bound of the rate distortion region when information sources satisfy the TS condition We further derive an expicit sufficient condtion for this outer bound to be tight In particuar we determine the rate sum part of the rate distortion region for the case where information sources satisfy the TS condition For some cass of Gaussian sources with the TS condition we derive an expicit recursive formua of this rate sum part Index Terms Mutitermina source coding many-hep-one probem Gaussian rate-distortion region CEO probem I INTROUCTION In muti-user source networks separate coding systems of correated information sources are significant from both theoretica and practica point of view The first fundamenta resut on those coding systems was obtained by Sepian and Wof They considered a separate source coding system of two correated information sources Those two sources are separatey encoded and sent to a singe destination where the decoder reconstruct the origina sources In the above source coding system we can consider the situation where the decoder wishes to reproduce one of two sources We ca this source the primary source In this case the remaining source that we ca the auxiiary source works as a partia side information at the decoder for the reconstruction of the primary source Wyner Ahswede and Körner 3 determined the admissibe rate region for this system the set that consists of a pair of transmission rates for which the primary source can be decoded with an arbitrary sma error probabiity We can naturay extend the system studied by Wyner Ahswede and Körner to the one where there are severa separatey encoded data of auxiiary sources serving as side Y Oohama is with the epartment of Information Science and Inteigent Systems University of Tokushima - Minami Josanjima-Cho Tokushima Japan informations at the decoder The determination of the admissibe rate region for this system is caed the many-hep-one probem In this sense Wyner Ahswede and Körner soved the so caed one-heps-one probem The many-hep-one probem has been known as a highy chaenging probem To date ony three soutions given by Körner and Marton 4 Gefand and Pinsker 5 and Oohama 8 are known Gefand and Pinsker 5 studied an interesting case of the many-hep-one probem They determined the admissibe rate region in the case where the auxiiary sources are conditionay independent if the primary source is given We hereafter say the above correation condition on the information sources the CI condition In Oohama 8 the author extended the many-hep-one probem studied by Gefand and Pinsker 5 to a continuous case He considered the many-hep-one probem for + correated memoryess Gaussian sources where auxiiary sources work as partia side information at the decoder for the reconstruction of the primary source The mean square error was adopted as a distortion criterion between the decoded output and the origina primary source output The rate distortion region was defined by the set of a transmission rates for which the average distortion can be upper bounded by a prescribed eve In 8 the author determined the rate distortion region when information sources satisfy the CI condition This resut contains the author s previous works for Gaussian one-heps-one probem 6 and Gaussian CEO probem 7 The probem sti remains open for Gaussian sources with genera correation Pandya et a 9 studied the genera case and derived an outer bound of the rate distortion region using some variant of bounding technique the author 6 used to prove the converse coding theorem for Gaussian one-heps-one probem However their bounding method was not sufficient to provide a tight resut Recenty in Oohama 0 the author extended the resut of 8 He considered a case of correation on Gaussian sources where + sources satisfy a kind of tree structure on their correation The author caed this tree structure of information sources the TS condition The TS condition contains the CI condition as a specia case In 0 the author derived an expicit outer bound of the rate distortion region for Gaussian sources satisfying TS condition Furthermore he had shown that for this outer bound coincides with the rate distortion region The author aso presented a sufficient condition for the outer bound to coincide with the rate distortion region Subsequenty Tavidar et a extended the TS condition to a binary Gauss Markov tree structure condition They studied a characterization of the rate distortion region for Gaussian

2 source with the binary tree structure In Oohama 0 the anaysis for matching condition of the inner and outer bounds was not sufficient In this paper we further investigate the matching condition of those two bounds and derive a condition much stronger than the matching condition in 0 We first derive an expicit outer bound of the rate distortion region when information sources satisfy the TS condition This outer bound is essentiay the same as the author s previous outer bound in 0 but it has a form more suitabe than the previous one for anaysis of a matching condition Using the derived outer bound we presented an expicit sufficient condition for the outer bound to coincide with the inner bound Furthermore we show that for the outer bound in this paper and that in 0 their rate sum parts coincide with the rate sum part of the inner bound Hence in the case where information sources satisfy the TS condition we estabish an expicit characterization of the rate sum part of the rate distortion region For some cass of Gaussian sources with the TS condition we derive an expicit recursive formua of the optima rate sum Our formua contains the resut of Oohama 7 for Gaussian CEO probem as a specia case II PROBEM STATEMENT AN PREVIOUS RESUTS In this section we state the probem formuation and previous resuts We first state some notations used throughout this paper et Φ Φ and A i i Φ be arbitrary sets Consider a random variabe A i i Φ taking vaues in A i We write n direct product of A i as A n i A i A i et a random vector consisting of n independent copies of the random variabe A i be denoted by A i A i A i A in We write an eement of A n i as a i a i a i a in et S be an arbitrary subset of Φ et A S and A S denote random vectors A i ) and A i ) respectivey Simiary et a S denote a vector a i ) When S k k + we aso use the notation A k for A S and use simiar notations for other vectors or random variabes When k we sometimes omit subscript Throughout this paper a ogarithms are taken to the natura A Forma Statement of the Probem et X i i 0 be correated zero mean Gaussian random variabes taking vaues in rea ines X i et Λ The CI condition Oohama 8 treated corresponds to the case where X X X are independent if X 0 is given In this paper we dea with the case where X X have some correation when X 0 is given et X 0t X t X t ) tbe a stationary memoryess mutipe Gaussian source For each t X 0t X t X t ) obeys the same distribution as X 0 X X ) The mutitermina source coding system treated in this paper is depicted in Fig For each i 0 the data sequence X i is separatey encoded to ϕ i X i ) by encoder function ϕ i The encoded data ϕ i X i ) i 0 are sent to the information processing center where the decoder observes them and outputs the estimation ˆX 0 of X 0 by n X 0 X X X 0 ϕ 0 X 0 ) ϕ 0 X ϕ X ) ϕ X ϕ X ) ϕ ψ ˆX0 Fig Communication system with side informations at the decoder using the decoder function ψ The encoder functions ϕ i i 0 are defined by and satisfy rate constraints ϕ i : X n i M i M i ) n og M i R i + δ ) where δ is an arbitrary prescribed positive number The decoder function ψ is defined by ψ : M 0 M M X n 0 3) enote by F n) δ R 0 R R ) the set that consists of a the + ) tupe of encoder and decoder functions ϕ 0 ϕ ϕ ψ) satisfying )-3) et dx ˆx) x ˆx) x ˆx) X0 be a square distortion measure For X 0 and its estimation ˆX 0 ψϕ 0 X 0 ) ϕ X ) ϕ X )) define the average distortion by X 0 ˆX 0 ) n EdX 0t n ˆX 0t ) t For a given > 0 the rate vector R 0 R R ) is admissibe if for any positive δ > 0 and any n with n n 0 δ) there exists ϕ 0 ϕ ϕ ψ) F n) δ R 0 R R ) such that X 0 ˆX 0 ) + δ et R ) denote the set of a the admissibe rate vector Our aim is to characterize R ) in an expicit form On the other hand we are often interested in a rate sum part of the rate distortion region R ) To examine this quantity define R sum R 0 ) R i min R 0R R ) R ) i To determine R sum R 0 ) in an expicit form is aso of our interest By the rate-distortion theory for singe Gaussian sources when R 0 og+ σ X 0 R R R 0 is admissibe Here og + a maxog a 0 Hence we have R ) R 0 R R ) : R 0 og+ σ X 0 R i 0 i Λ Throughout this paper we assume that σ X 0 and R 0 < og σ X 0

3 3 X 0 Z Y Y X N Z Y Y N Z3 Y 3 Y 3 X X 3 N 3 Fig TS condition in the case of 4 X 0 X 3 N 3 N4 X 4 Z 0 X 0 X 0 N Z 0 X 0 X 0 X N Z3 0 X 0 X 0 X Fig 3 CI condition in the case of 4 B Tree Structure of Gaussian Sources N4 X 4 In this subsection we expain the tree structure of Gaussian source which is an important cass of correation Consider the case where the + random variabes X 0 X X satisfy the foowing correations Y 0 X 0 Y Y + Z 4) X Y + N X Y N Z where Z i i Λ are independent Gaussian random variabes with mean 0 and variance σz i and N i i are independent Gaussian random variabes with mean 0 and variance σn i We assume that Z is independent of X 0 and that N is independent of X 0 and Z We can see that the above X 0 X X ) has a kind of tree structurets) We say that the source X 0 X X ) satisfies the TS condition when it satisfies 4) The TS condition contains the CI condition as a specia case by etting σ Zi i be zero et S be an arbitrary subset of Λ The TS condition is equivaent to the condition that for S Λ the random variabes X S X 0 Z ) X S c form Markov chains X S X 0 Z ) X S c in this order The TS and CI conditions in the case of 4 are shown in Fig and 3 respectivey C Previous Resuts In this subsection we state the previous resuts on the determination probem of R ) et U i i 0 be random variabes taking vaues in rea ines U i For S Λ define G) U 0 U ) : U 0 U ) is a Gaussian random vector that satisfies U X X 0 U 0 U S X S 0 X S c U S c for any S Λ and EX 0 ψu 0 U ) for some inear mapping ψ : U 0 U X 0 where S c Λ S et ) i π π) πi) π) be an arbitrary permutation on Λ and Π be a set of a permutations on Λ For S Λ we set πs) πi) efine subsets S i i of Λ by S i i i + Set R π ) R 0 R R ) : There exists a random vector U 0 U ) G) such that R 0 IX 0 ; U 0 U ) R πi) IX πi) ; U πi) U πs c i )) for i R in) ) conv R in) π ) π Π where conva denotes a convex hu of the set A Then we have the foowing Theorem Oohama 8): For Gaussian sources with genera correation R in) ) R ) For Gaussian sources with the CI condition the inner bound R in) ) is tight that is R in) ) R ) in) The above inner bound R ) can be regarded as a variant of the inner bound which is we known as the inner bound of Berger and Tung 3 Theorem contains the soution that Oohama 6 obtained to the one-heps-one probem for Gaussian sources as a specia case When R 0 0 the second resut of Theorem has some impications for the Gaussian CEO probem studied by Viswanathan and Berger 4 and Oohama 7 and source coding probem for mutitermina communication systems with a remote source investigated by Yamamoto and Itoh 5 and Fynn and Gray 6 The notion of TS condition for Gaussian sources was first introduced by Oohama 0 Tavidar et a extended the TS condition to a binary Gauss Markov tree structure condition They studied a characterization of the rate distortion region for Gaussian sources with a binary tree structure III RESUTS ON THE RATE ISTORTION REGION In this section we state our main resuts on inner and outer bounds of R ) in the case where X 0 X X ) satisfies the TS condition

4 4 A efinition of Functions and their Properties In this subsection we define severa functions which are necessary to describe our resuts and present their properties et r i i Λ be nonnegative numbers efine the sequence of nonnegative functions f r ) f 0r ) by the foowing recursion: f r ) e r σ N + e r σ N f r ) f + r + ) +σ Z + f + r + ) + e r σ N f 0 r ) f r ) +σ Z f r ) Next we define the sequence of nonnegative functions g r 0 ) 0 g r 0 r ) by the foowing recursion: g 0 r 0 ) e r 0 σx 0 g r 0 ) g 0r 0) σ Z g 0r 0) g + r 0 r ) + 6) g r 0r ) σ e r ) N + σ g Z r + 0r ) σ N e r ) where a + maxa 0 et B ) be the set of a nonnegative vectors r0 that satisfy f 0 r ) g 0 r 0 ) e r 0 σ X 0 et B ) be the boundary of B ) that is the set of a nonnegative vectors r 0 that satisfy f 0 r ) g 0 r 0 ) e r 0 σ X 0 We can easiy show that the functions we have defined satisfy the foowing property Property : a) For each i Λ f 0 r ) is a monotone increasing function of r i For each and for each i + f r ) is a monotone increasing function of r i b) For each and for each i 0 g r 0 r ) is a monotone decreasing function of r i c) If r 0 B ) then for 0 g r 0 r ) f r ) In the above inequaities the equaities simutaneousy hod if and ony if r 0 B ) efine Fr ) + σ Z f r ) G r 0 r ) + σ Z g r 0 r ) 5) For S Λ define f 0 r S ) f 0 r ) rs c0 Fr S) Fr ) rs c0 We can easiy show that the functions Fr ) and G r 0 r ) satisfy the foowing property Property : a) For each i S Fr S ) is a monotone increasing function of r i b) For each i 0 G r 0 r ) is a monotone decreasing function of r i c) If r 0 B ) then G r 0 r ) Fr ) The equaity hods if and ony if r 0 B ) For > 0 r i 0 i Λ and S Λ define J S r 0 r r S r S c) og+ Gr 0r ) Fr S c) K S r S r S c) og Fr ) Fr S c) +σ X 0 f 0r ) σ X 0 e r0 +σ f X 0 0r S c) +σ f X 0 0r S c) e ri e ri We can show that for S Λ K S r S r S c) and J S r 0 r r S r S c) satisfy the foowing two properties Property 3: a) If r 0 B ) then for any S Λ J S r 0 r r S r S c) K S r S r S c) The equaity hods when r0 B ) b) Suppose that r B ) If r rs0 sti beongs to B ) then J S r 0 r r S r S c) K rs0 Sr S r S c) rs0 0 Property 4: Fix r B ) For S Λ set ρ S ρ S r S r S c) J S r 0 r r S r S c) By definition it is obvious that ρ S S Λ are nonnegative We can show that ρ ρ S S Λ satisfies the foowings: a) ρ 0 b) ρ A ρ B for A B Λ c) ρ A + ρ B ρ A B + ρ A B In genera Λ ρ) is caed a co-poymatroid if the nonnegative function ρ on Λ satisfies the above three properties Simiary we set ρ S ρ S r S r S c) K S r S r S c) ρ ρ S S Λ Then Λ ρ) aso has the same three properties as those of Λ ρ) and becomes a co-poymatroid

5 5 B Resuts In this subsection we present our resuts on inner and outer bounds of R ) In the previous work 0 we derived an outer bound of R ) We denote this outer bound by ˆR out) Set ) According to 0 ˆRout) ) is given by ˆR out) ) R 0 R ) : There exists a nonnegative vector r 0 r ) such that R 0 r 0 og+ R i r i for any i Λ R 0 + R i og+ + i r i for any S Λ σx 0 +σ f X 0 0r ) Gr 0r )σ X 0 Fr S c) +σx f 0 0r S c) R out) r0 ) R 0 R R ) : R 0 r 0 R i J S r0 r r S r S c) for any S Λ R in) r 0 ) R 0 R R ) : R 0 r 0 R i K S r S r S c) R out) ) R in) ) for any S Λ r0 ) r 0 B) R out) r 0 B) R in) r 0 ) Our main resut is as foows Theorem : For Gaussian sources with the TS condition R in) in) ) R ) R ) ˆR out) ) R out) ) Proof of this theorem wi be given in Section V The out) incusion R ) ˆR ) and an outine of proof of this incusion was given in Oohama 0 Furthermore by in) Theorem we have R ) R ) Hence it suffices to out) show ˆR ) R out) ) and R in) in) ) R ) to prove Theorem Proofs of those two incusions wi be given in Section V We can directy prove R ) R out) ) in a manner simiar to that of Oohama 0 For the detai of the direct proof of R ) R out) ) see Appendix B An essentia difference between R out) ) and R in) ) is the difference between J S r 0 r r S r S c) in the definition of R out) of R in) R out) ) and K S r S r S c) in the definition ) By Property 3 part a) and the definitions of r0 ) and R in) r 0 ) if r0 B ) then R out) r0 ) R in) r 0 ) This gap suggests a possibiity that in some cases those two bounds match In the foowing we present a sufficient condition for R out) ) R in) ) We consider the foowing condition on G r 0 r ) Condition: For each e ri G r 0 r ) is a monotone increasing function of r We ca the above condition the MI condition The foowing is our main resut on a matching condition on inner and outer bounds emma : For Gaussian sources with the TS condition if G r 0 r ) satisfies the MI condition then R out) ) R in) ) Proof of this emma is given in Section V Note that when or σ Z 0 for 3 we have G r 0 r ) + σ Z g r 0 ) which satisfies the MI condition Combining emma and Theorem we estabish the foowing Theorem 3: For Gaussian sources with the TS condition R in) ) R ) ˆR out) ) R out) ) Furthermore if G r 0 r ) satisfies the MI condition then ˆR out) R in) ) R ) ) R out) ) out) In Oohama 0 the equaity R ) ˆR ) was stated without compete proof We can see that this equaity can be obtained by Theorem emma and the fact that the MI condition hods for Next we present a sufficient condition for G r 0 r ) to satisfy the MI condition et fj j positive numbers defined by the foowing recursion: f σ N + σ N f f + +σ Z + f + + σ N be a sequence of By definition it is obvious that f r ) f Then we have the foowing proposition Proposition : If k 7) σ ) Z k+ k ) + σ σ Z k+ fk+ + σz j fj 8) N j+ hod for then G r 0 r ) satisfies the MI condition Proof of this proposition wi be given in Appendix A It can be seen from this proposition that for 3 the MI condition hods for reativey sma vaues of σ Z In particuar when 3 the sufficient condition given by 8) is + σz σz σ N σ N + σ N 3 )

6 6 X 0 Z Y Y N Z Y Y X X N N3 X 3 Fig 4 TS condition in the case of 3 Soving the above inequaity with respect to σ Z we have σ Z + + 4σ N σ N + σ N 3 ) σ N The TS condition in the case of 3 is shown in Fig 4 IV RATE SUM PART OF THE RATE ISTORTION REGION In this section we state our resut on the rate sum part of R ) Set R ) sum R 0) R u) sum R 0) ˆR ) min J Λ R 0 r r ) r :f 0r ) g 0R 0) min K Λ r ) r :f 0r ) g 0R 0) ˆR out) et sum R 0) be the minimum rate sum for ) Then it immediatey foows from Theorem that we have the foowing coroary Coroary : For Gaussian sources with the TS condition R ) sum R 0) ˆR ) sum R 0) R sum R 0 ) R u) sum R 0) On the other hand we have the foowing emma emma : For Gaussian sources with the TS condition we have R ) sum R 0) R u) sum R 0) Proof of this emma wi be given in Section V Combining Coroary and emma we have the foowing Theorem 4: For Gaussian sources with the TS condition R sum R 0 ) R u) sum R 0) ˆR ) sum R 0) R ) sum R 0) min r + r :f 0r ) og Fr ) g 0R 0) R 0 + og σ X 0 The optima rate sum R sum R 0 ) has a form of optimization probem We dea with this optimization probem in a case where σ N σ for and σ Z ǫ σ for In this case we sove the optimization probem by deriving a parametric form of R sum R 0 ) In the above case of identica variances the recursion 5) is f + f + ǫ + σ + ) e r f + σ for 9) The optimization probem presenting R sum R 0 ) is R sum R 0 ) min r :f 0r ) g 0R 0) R 0 + og σ X 0 For we set r + og + ǫ σ f r )) α σ f r ) +ǫ σ f r ) for α e r for 0) By the above transformation we transform the variabe r into α From 0) we have f σ α for ) ǫ α Since f 0 α must satisfy 0 α < ǫ From 9) and ) we have e r α + α ǫ α Since r 0 for α must satisfy α α ǫ α < α ǫ α for α ǫ α < α < for ) et ǫ and et 0 ǫ ) be a direct product of the semi-open intervas 0 ǫ ) et A be a set of a dimensiona vectors α 0 ǫ ) that satisfy ) Using α R sum R 0 ) is rewritten as R sum R 0 ) min α ) Aα) α σ g 0R 0) R 0 + og σ X 0 og + og ǫ α ) ) α + α + ǫ α + og α ) where A α ) α : α α α ) A To sove the above optimization probem set ζ ζα ) ) α og + α + ǫ α + og ǫ α ) + og α ) Then we have the foowing emma

7 7 emma 3: For α Aα ) ζα ) is stricty concave with respect to α Proof of this emma wi be given in Appendix C It can be seen from this emma that if we can find θ satisfying ζ α θ 0 and θ A σ g 0 R 0 )) this θ is the unique vector which attains R sum R 0 ) We sha give such θ in an expicit form of recursion et ω 0 ) efine the sequence of functions θ ω) by the foowing recursion: θ ω) ω θ ω) θ ω) for θ ω) + ǫ θ ω) ω + ǫ ω θ ω) ǫ )θ + ω) ǫ +ǫ +θ + ω)) + ǫ θ ω) ǫ )θ + ω) ǫ +ǫ +θ + ω)) 3) Then we have the foowing emma emma 4: The sequence of functions θ ω) defined by 3) satisfies the foowings a) b) 0 θ ω) θ ω) ǫ θ ω) θ ω) ǫ θ ω) θ ω) ǫ < θ θ ω) ω) < 0 θ ω) θ ω) ǫ θ ω) ǫ +ǫ θ ω) ǫ θ ω) < θ ω) < ǫ For ǫ + +ǫ + +ǫ ǫ + θ ω) θ ω) ǫ θ ω) θ ω) ǫ θ ω) < θ ω) < ǫ ǫ +ǫ The above 4)-6) impy that θ ω) A θ ω)) ζ α θ ω) 0 4) 5) 6) c) For each θ ω) is differentiabe with respect to ω 0 ) and satisfies the foowing: dθ dω ) +ǫ ω j+ + ) +) +ǫ ω+) +ǫ j θ jω)+) j+ +ǫ j θ jω)+) > 0 This impies that for each the mapping ω 0 ) θ ω) is an injection Proof of this emma is given in Appendix From this emma we obtain the foowing theorem Theorem 5: et θ ω) be a sequence of functions defined by 3) Then we have the foowing parametric form of R sum R 0 ) with the parameter ω 0 ): σ g 0 R 0 ) σ e R 0 θ σ ω) X 0 R sum R 0 ) ) og R 0 + og σ X 0 ) θ ω) ǫ θ ω) + θ +ω) + og ǫ θ ω)) + og ω) When ǫ 0 for the recursion 3) becomes the foowing: θ ω) ω θ ω) ω θ ω) θ ω) θ + ω) 7) for Soving 7) we obtain θ ω) +)ω The parametric form of R sum R 0 ) becomes σ g 0 R 0 ) θ ω) ω R sum R 0 ) ) og ω) 8) R 0 + og σ X 0 From 8) we have R sum R 0 ) ) ) og σ g 0 R 0 ) R 0 + og σ X 0 9) In particuar by etting R 0 0 and in 9) we have im R sum 0) σ g 0 0) + og σ X 0 σ σ X0 σx 0 + og σ X 0 The above formua coincides with the rate distortion function for the quadratic Gaussian CEO probem obtained by Oohama 7 Hence our soution to R sum R 0 ) incudes the previous resut on the Gaussian CEO probem as a specia case V PROOFS OF THE RESUTS In this section we prove Theorem and emma stated in Section III and prove emma stated in Section IV A erivation of the Outer Bound out) In this subsection we prove ˆR ) R out) ) stated in Theorem Proof of ˆRout) ) R out) ): Set Ĵ S r 0 r r S r S c R 0 ) og+ Fr S c) Gr 0r )σ X 0 +σx f 0 0r S c) + + r i R 0 i

8 8 We first observe that Ĵ S r 0 r r S r S c r 0 ) og + Fr S c) Gr 0r og )σx 0 Fr S c) +σ f X 0 0r S c) Gr 0r )σ X 0 +σx f 0 0r S c) r i r 0 i + r i r 0 i J S r 0 r r S r S c) 0) Then we have the foowing a) ˆR out) ) R 0 R ) : There exists a nonnegative vector r 0 r ) such that R 0 r 0 og+ σ X 0 +σ f X 0 0r ) R i ĴS r 0 r r S r S c R 0 ) for any S Λ b) R 0 R ) : There exists a nonnegative vector r such that R 0 og+ σ X 0 +σx f 0 0r ) R i ĴS R 0 r r S r S c R 0 ) for any S Λ R 0 R ) : There exists a nonnegative vector r 0 r ) such that R 0 r 0 og σx 0 +σ f X 0 0r ) R i ĴS r 0 r r S r S c r 0 ) for any S Λ c) R 0 R ) : There exists a nonnegative vector r 0 r ) such that R 0 r 0 og σ X 0 +σ f X 0 0r ) R i J S r 0 r r S r S c) for any S Λ R out) ) Step a) foows from the definition of Ĵ S R 0 r r S r S c R 0 ) and the nonnegative property of R Step b) foows from that Ĵ S r 0 r r S r S c R 0 ) is a monotone decreasing function of r 0 Step c) foows from 0) Thus ˆR out) ) R out) ) is proved B erivation of the Inner Bound In this subsection we prove R in) ) R ) stated in Theorem We first derive a preiminary resut on a form of R in) ) Fix R 0 r 0 and set R in) r 0 R 0 ) R R ) : R 0 R R ) R in) r 0 ) et Λ ρ) ρ ρ S r S r S c) S Λ be a co-poymatroid defined in Property 4 Expression of R in) r 0 R 0 ) using Λ ρ) is R in) r 0 R 0) R R ) : R i ρ S r S r S c) for any S Λ The set R in) r 0 R 0 ) forms a kind of poytope which is caed a co-poymatroida poytope in the terminoogy of matroid theory It is we known as a property of this kind of poytope that the poytope R in) r 0 R 0) consists of! end-points whose components are given by R πi) ρ πi) π) r πi) π) r π) πi ) ) ρ πi+) π) r πi+) π) r π) πi) ) for i R π) ρ π) r π) r π) π ) ) ) where ) i π Π π) πi) π) is an arbitrary permutation on Λ For each π Π and r0 B ) et R in) π r 0 ) be the set of nonnegative vectors R 0 R R ) satisfying R 0 r 0 R πi) ρ πi) π) r πi) π) r π) πi ) ) ρ πi+) π) r πi+) π) r π) πi) ) for i R π) ρ π) r π) r π) π ) ) Then we have R in) r 0 ) conv π Π R in) π r 0 ) ) Proof of R in) in) ) R ): Fix π Π and r0 B ) arbitrary By ) it suffices to show that for r0 B ) R in) π r in) 0 ) R π ) to prove Rin) ) R in) ) et V i i 0 Λ be independent Gaussian random variabes with mean 0 and variance σv i Suppose that V0 is independent of X0 efine the Gaussian random variabes U i i 0 Λ by U i Xi + V i i 0 Λ From the above definition it is obvious that U X X 0 U 0 U S X S 0 X S c U S c for any S Λ 3)

9 9 For given r i 0 i S and > 0 set σ V i er i when σ N i r i > 0 When r i 0 we choose U i so that U i takes the constant vaue zero efine the sequence of random variabes Ω 0 by Ω e r σ N U + e r σ N U Ω +σ Z + f + r + ) Ω + + e r σ N U for Ω 0 +σz f r ) Ω Note that Ω 0 Ω 0 U ) is a inear function of U Then by an eementary computation we have X f σ σ 0 r ) X 0 V 0 σ V 0 U 0 + Ω 0 U ) 4) +Ñ0 5) where Ñ0 is a zero mean Gaussian random variabe with variance + + f σx σ 0 r ) 0 V 0 Ñ 0 is independent of U 0 U ) Since r 0 B ) we have We put Then from 6) and 7) we have e r 0 σ X 0 f 0 r ) 6) + σx 0 σ V 0 e r 0 7) σ V 0 + f 0 r ) σ X 0 + e r 0 + f 0 r ) 8) Based on 5) 7) and 8) define the inear function ψ of U 0 U ) by ψu 0 U ) + e r 0 σ + f 0 r ) X 0 e r 0 U 0 + Ω 0 U ) Then we obtain E X 0 ψu 0 U ) Var Ñ0 + e r 0 σ + f 0 r ) 9) X 0 From 3) and 9) we have U 0 U ) G) By simpe computations we can show that r 0 IX 0 ; U 0 U ) r i IX i ; U i X 0 Y ) for any i Λ og F S r S ) + σx 0 f 0 r S ) 30) IX 0 Y ; U S ) for any S Λ Using 3) and 30) the + inequaities of ) are rewritten as R 0 IX 0 ; U 0 U ) R πi) IX 0 Y ; U πsi) U πs c i )) +IX πi) ; U πi) X 0 Y ) IX 0 Y ; U πsi+) U πs c i+ )) IX 0 Y ; U πi) ; U πs c i )) +IX πi) ; U πi) X 0 Y U πs c i )) IX 0 Y X πi) ; U πi) U πs c i )) IX πi) ; U πi) U πs c i )) for i in) Thus we concude that R 0 R π) R π) ) R π ) C Proofs of emmas and In this subsection we prove emmas and We first present a preiminary observation on R out) ) Fix R 0 r 0 arbitrary and set R out) r 0 R 0 ) R R ) : R 0 R R ) R out) r 0 ) et Λ ρ) ρ ρ S r S r S c) S Λ be a co-poymatroid defined in Property 4 Expression of R out) 0 r0 R 0) using Λ ρ) is R out) r 0 R 0 ) R R ) : R i ρ S r S r S c) for any S Λ The set R out) r0 R 0) forms a co-poymatroida poytope The poytope R out) r0 R 0) consists of! end-points whose components are given by R πi) ρ πi) π) r πi) π) r π) πi ) ) ρ πi+) π) r πi+) π) r π) πi) ) for i R π) ρ π) r π) r π) π ) ) 3) where ) i π Π π) πi) π) For each π Π and set B π ) r0 : r0 B ) and r πi) 0 for i + B π ) r0 : r0 B ) and r πi) 0 for i + In particuar when π is the identity map we omit π to write B ) and B ) By Property 3 when r0 B π) the

10 0 end-point given by 3) becomes R πi) ρ πi) π) r πi) π) r π) πi ) ) ρ πi+) π) r πi+) π) r π) πi) ) for i R π) ρ π) r π) r π) π ) ) R πi) 0 for i + 3) Next we present a emma on a property of G r 0 r ) emma 5: For r0 B ) G r 0 r ) is computed as G r 0 r ) r σ Zk g k r 0 r k ) k Proof: By Property part c) for + k 0 g k r 0 r k ) fr k ) 0 Hence the resut of emma 5 foows Proof of emma : Fix π Π and r0 B ) arbitrary et R 0 R ) be a nonnegative rate vector such that R 0 r 0 and components of R satisfy 3) To prove emma it suffices to show that this nonnegative vector beongs to R in) ) For we prove the caim that under the MI condition if r0 B π ) then the rate vector R 0 R ) satisfying R 0 r 0 and 3) beongs to R in) ) We prove this caim by induction with respect to When from 3) we have R π) ρ π) r π) ) 33) R πi) 0 for i The function ρ π) r π) ) is computed as ρ π) r π) ) J π) r 0 r r π) r π) c) rπ) c0 Gr0r ) rπ) og+ c0 σ X 0 e r0e r π) 34) By the above form of ρ π) r π) ) and σ X 0 e r 0 σ X 0 e R0 > ρ π) r π) ) is positive Since r 0 B π ) we can decrease r π) keeping r 0 B π ) so that it arrives at r π) 0 or a positive r π) satisfying r 0 r π) r π) c) r 0 rπ) 0 0) B π ) 35) et R 0 Rπ) R π) ) be a rate vector corresponding to r 0 rπ) r π) c) If r π) 0 then by Property 3 part b) ρ π) r π) ) must be zero This contradicts the fact that ρ π) r π) ) is positive Therefore rπ) must be positive Then from 35) we have R 0 R π) R π) ) R 0 Rπ) 0 0) R in) ) On the other hand by emma 5 we have G r 0 r ) rπ) c0 G r 0 r ) r π)+ 0rπ) 0 π) k + σ Z g k r 0 r k ) r π) 0 36) From 34) and 36) we can see that G r 0 r ) rπ) c0 does not depend on r π) This impies that ρ π) r π) ) is a monotone increasing function of r π) Then we have R π) Rπ) Hence we have R 0 R π) R π) ) R 0 R π) 0 0) R in) ) Thus the caim hods for We assume that the caim hods for Since f 0 r0 ) is a monotone increasing function of r π) on B π ) we can decrease r π) keeping r0 B π ) so that it arrives at rπ) 0 or a positive r π) satisfying r 0 rπ) r π) c) B π) 37) et R 0 Rπ) R π) ) be a rate vector corresponding to r 0 rπ) r π) c) By Property 4 part b) and the MI condition the functions ρ πi) π) r πi) π) r π) πi ) ) ρ πi+) π) r πi+) π) r π) πi) ) for i ρ π) r π) r π) π ) ) appearing in the right members of 3) are monotone increasing functions of r π) Then from 3) we have R πi) Rπi) for i R πi) Rπi) 38) 0 for i + When r π) 0 we have r 0 r π) r π) c) B π ) Then by induction hypothesis we have R 0 Rπ) R π) ) Rin) ) When rπ) > 0 from 37) we have R 0 Rπ) R π) ) Rin) ) Hence by 38) we have R 0 R π) R π) ) R 0 R π) R π) 0 0) R in) ) Thus the caim hods for This competes the proof of emma Proof of emma : For R 0 > 0 and for set B R 0 ) r : R 0 r ) B ) B R 0 ) r : R 0 r ) B )

11 We first observe that R ) sum R 0) min R u) min J Λ R 0 r r ) r r B R 0) + 0 sum R 0) min min K Λ r ) r B R 0) We compute J Λ R 0 r r ) r By emma 5 for 0 + r B R 0 ) G R 0 r ) r σ Z g k R 0 r k ) k From the above formua we can see that for r B R 0 ) G R 0 r ) r + 0 is a function of r We denote this function by G R 0 r ) that is G R 0 r ) Then for r B R 0 ) + σ Z g k R 0 r k ) k J Λ R 0 r r ) r + 0 og+ G R 0 r ) σ X 0 e R0 e ri 39) i We denote the right member of 39) by J Λ R 0 r r ) Using this function R ) sum R 0) can be written as R ) sum R 0) min min J Λ R 0 r r ) r B R 0) Note here that J Λ R 0 r r ) is a monotone increasing function of r To prove R ) sum R 0) R u) sum R 0) it suffices to show that for min J Λ R 0 r r ) min K Λ r ) r B R 0) r B R 0) We prove this caim by induction with respect to When the function J Λ R 0 r ) is computed as J Λ R 0 r ) +σ og+ Z g R 0)σ X 0 e R 0e r og+ σ X 0 e R 0e r σ Z g 0R 0) Since σ X 0 e R 0 > J Λ R 0 r ) is positive Since J Λ R 0 r ) is a monotone increasing function of r the minimum of this function is attained by r 0 or a positive r satisfying r B R 0 ) If r 0 then by Property 3 part b) J Λ R 0 r ) must be zero This contradicts that J Λ R 0 r ) is positive Therefore r must be positive Then by r B R 0 ) we have J Λ R 0 r ) J Λ R 0 r ) K Λ r ) min r B R 0) K Λr ) Thus the caim hods for We assume that the caim hods for Since J Λ R 0 r r ) is a monotone increasing function of r the minimum of this function is attained by r 0 or a positive r satisfying r r ) B R 0 ) When r 0 we have r B R 0 ) and J Λ R 0 r r ) J Λ R 0 r r r ) 40) Computing J Λ R 0 r r r ) we obtain J Λ R 0 r r r ) J Λ R 0 r r ) r 0 og+ G R 0 r ) σ X 0 e R0 e ri i J Λ R 0 r r ) 4) Combining 40) and 4) we have J Λ R 0 r r ) J Λ R 0 r r ) 4) On the other hand by induction hypothesis we have J Λ R 0 r r ) Combining 4) and 43) we have J Λ R 0 r r ) When r > 0 we have min K Λ r ) 43) r B R 0) min K Λ r ) r B R 0) min K Λ r ) r B R 0) J Λ R 0 r r ) J Λ R 0 r r r ) K Λ r r ) min K Λ r ) r B R 0) where the second equaity foows from r r ) B R 0 ) Thus the caim hods for competing the proof VI CONCUSIONS We have considered the Gaussian many-hep-one probem and given a partia soution to this probem by deriving expicit outer bound of the rate distortion region for the case where information sources satisfy the TS condition Furthermore we estabished a sufficient condition under which this outer bound is tight We have determined the rate sum part of the rate distortion region for the case where information sources satisfy the TS condition For the case that information sources do not satisfy the TS condition we can not derive an outer bound having a simiar form of R out) ) since the proof of the converse coding theorem depends heaviy on this property of information sources Hence the compete soution is sti acking for Gaussian information sources with genera correation

12 og G r 0 r ) can be rewritten as A Proof of Proposition APPENIX In this appendix we prove Proposition To prove this proposition we give some preparations For 0 we set η η r 0 r ) g0 r 0 ) for 0 g r 0 r ) σ N e r ) for For and a < σ Z define τ a) Then η 0 Fix a < σ Z k+ and set ) a+ σ a Z + σ e r N satisfies the foowing: η r 0 r ) τ η r 0 r ) ) for 44) p k a) sup p :og σ Zk+ a + pb a) σ b Z + k+ for any b < σ Z k+ By a simpe computation we have σz k+ p k a) σ a for 0 a < σ Z k+ Z k+ 0 for a < 0 σ Z k+ σ Z k+ a for a < σ Z k+ 45) Fix a < and set σz j q j a) sup q :τ j b) τ j a) qb a) for any b < σ Z j By a simpe computation we have for 0 a < σ q j a) a) Z σ j Z j 0 for a < 0 σ Z j a) for a < σ Z j 46) Proof of Proposition : et be a set of integers such that η r 0 r ) takes a positive vaue for some r0 B ) From 44) there exists a unique integer such that 0 Using η and og G r 0 r ) og + σz k g r 0 r k ) k k k0 k0 og og og σz η k r k 0r k ) + σ η Z k r k+ 0r k ) + σ η Z k r k+ 0r 47) k ) + Fix nonnegative vector r For each s r et Gs ) be a function obtained by repacing r in G r 0 r ) with s that is Gs ) G r 0 r s r + ) It is obvious that when s r Gr ) G r 0 r r r + ) G r 0 r ) By Property part b) we have Gs ) Gr ) for For each s k r k k et η k s ) be a function obtained by repacing r in η k r 0 r k ) with s that is η k s ) η k r 0 r s r k + ) It is obvious that when s r η k r ) η k r 0 r r r k + ) η k r 0 r k ) By Property part b) we have η k s ) η k r ) for k For each we evauate an upper bound of og Gs ) og Gr ) Using 47) we have og Gs ) Gr ) k0 k σ Z η k r ) + og k+ σz η k s ) + k+ σ Z η k r ) + og k+ σ η Z k s 48) ) + k+ By definition of p k ) we have σ Z η k r ) + og k+ σ η Z k s p ) + k η k r ))η k s ) η k r ) k+ σ Z k+ σ Z k+ η k r ) η ks ) η k r ) 49) where the ast inequaity foows from η k s ) η k r ) and 45) From 48) and 49) we have og Gs ) Gr ) σz k+ σ η Z k r ) η ks ) η k r )) 50) k+ k By definition of q j ) and 44) for + j k we have η j s ) η j r ) q j η j r ))η j s ) η j r ) ) η j s ) η j r ) 5) σ η Z j j r )

13 3 where the ast inequaity foows from η j s ) η j r ) and 46) Using 5) iterativey for + j k we obtain η k s ) η k r ) η s ) η r )) k j+ σ Zj η j ) 5) Observe that η s ) η r ) σ e s e r N From 5) and 53) we have η k s ) η k r ) e r σ N s r ) σ N s r ) k j+ From 50) and 54) we have og Gs ) Gr ) s r ) k e r σ N s r ) 53) k j+ σ Zj η j ) σ Zj η j ) 54) σ Z k+ σ N σ Z k+ η k k j+ σ Zj η j ) 55) By Property part b) and the definition of η j we have η j f j σ N j e rj ) from which we have f j+ +σ Z j+ f j+ σ Z j+ η j + σ Z j+ f j+ + σ Z j+ f j+ 56) From 55) and 56) we have If og Gs ) Gr ) s r ) k k σ ) Z k+ + σ σ Z k+ fk N k j+ + σ Z j f j ) 57) σ ) Z k+ k ) + σ σ Z k+ fk+ + σz j fj 58) N j+ hod for then by 57) we have or equivaent to og Gs ) Gr ) s r ) s + og Gs ) r + og Gr ) for Hence 58) is a sufficient condition for the MI condition B Proof of R ) R out) ) In this appendix we prove R ) R out) ) stated in Theorem We first present a emma necessary for the proof of this incusion emma 6: IX 0 ; ˆX 0 ) n ) og σ X 0 X 0 ˆX 0) Proof: See the proof of emma in Oohama 7 Next we present an important emma which is a mathematica core of the converse coding theorem et the encoded outputs of X i i 0 by encoder functions ϕ i be denoted by ϕ i X i ) W i Set r 0 n IX 0; W 0 W ) r i n IX i; W i Y ) n IX i; W i Y i ) for i n IX ; W Y ) for i ξ σ X 0 e n IX0;W0W ) Then we have the foowing emma emma 7: IX 0 ; W ) n og + σ X 0 f 0 r ) For we have n og + σz g r 0 r ξ) IY ; W Y ) n og + σz f r ) From the above emma we immediatey obtain the foowing emma 8: IX 0 ; W S ) n og + σ X 0 f 0 r S ) IY ; W S X 0 ) n og Fr S) S Λ IY ; W X 0 ) n og Gξ r 0 r ) We prove R ) R out) ) by emmas 6 and 8 and standard arguments for the proof of the converse coding theorem Proof of R ) R out) ): We first observe that by virtue of the TS condition W S X S X 0 Z ) X S c W S c 59) hod for any subset S of Λ Assume R 0 R R ) R ) Then for any δ > 0 there exists an integer n 0 δ) such that for n n 0 δ) and for i Λ we obtain the foowing chain of inequaities: nr 0 + δ) og M 0 HW 0 ) HW 0 W ) IX 0 ; W 0 W ) nr 0 60)

14 4 Furthermore for any subset S Λ we obtain nr 0 + nr i + δ) IX 0 ; W 0 W ) + HW i ) HW 0 W ) + HW i ) HW 0 W S W S c) + HW S W S c) HW 0 W S W S c) IX 0 Z ; W 0 W S W S c) +HW 0 W S W S cx 0 Z ) a) IX 0 Z ; W 0 W S W S c) + HW i X 0 Z ) IX 0 Y ; W 0 W S W S c) + IX i ; W i Y ) 6) Step a) foows from 59) On the other hand by emma 6 we have for n n 0 δ) IX 0 ; W 0 W ) n ) og σ X 0 ξ IX 0 ; ˆX 0 ) n ) og σ X 0 +δ which together with 60) 6) and emma 8 yieds the foowing ower bounds of IX 0 ; W 0 W ) and IX 0 Y ; W 0 W S W S c): IX 0 ; W 0 W ) IX 0 ; W 0 W ) IX 0 ; W ) n og σ X 0 +σx f 0 0r ) ξ n og σx 0 6) +σ X 0 f 0r ) +δ) IX 0 Y ; W 0 W S W S c) IX 0 Y ; W 0 W S W S c) IX 0 Y ; W S c) IX 0 ; W 0 W ) + IY ; W X 0 ) IX 0 ; W S c) IY ; W S c X 0 ) n og σ X 0 Gξr 0r ) Fr S c) +σ f X 0 0r S c) ξ n og σ X 0 G+δr 0r ) 63) Fr S c) +σ X 0 f 0r S c) From 6) and 63) we have R i + δ) og + +δ) σ X 0 G+δr 0r ) Fr S c)+δ) +σ X 0 f 0r S c) r i r 0 64) Note here that R i + δ) are nonnegative Hence from 60) 6) and 64) we obtain R 0 + δ r 0 og σx 0 65) +σ X 0 f 0r ) +δ) and for S Λ R i + δ) J S + δ r 0 r r S r S c) The inequaity 65) impies that r0 B + δ) Thus by etting δ 0 we obtain R 0 R R ) R out) ) Finay we prove emma 7 For n dimensiona random vector U with density et hu) be a differentia entropy of U The foowing two emmas are some variants of the entropy power inequaity emma 9: et U i i 3 be n dimensiona random vectors with densities and et T be a random variabe taking vaues in a finite set We assume that U 3 is independent of U U and T Then we have πe e n hu+u3 UT) πe e n hu UT) + πe e n hu3) emma 0: et U i i 3 be n random vectors with densities et T T be random variabes taking vaues in finite sets We assume that those five random variabes form a Markov chain T U U 3 U T in this order Then we have πe e n hu+u U3TT) πe e n hu U3T) + πe e n hu U3T) Proof of emma 7: efine the sequence of n dimensiona random vectors S by S σ N X + σ Z + Y + 66) By an eementary computation we obtain X 0 σˆn 0 Y σ + ˆN 0 Z Y σˆn Y σz + σ ˆN S + ˆN 67) where ˆN 0 is an n dimensiona random vector whose components are n independent copies of a Gaussian random variabe with mean 0 and variance σ ˆN Nˆ 0 is independent of Y For each ˆN is independent of Y and S The variance σ ˆN 0 have the foowing form: Set + σ ˆN σ σ X 0 0 Z + + σ ˆN σ σ Z N σ Z + λ 0 πe e n hx0 W ) µ 0 πe e n hy W ) µ 0 πe e n hy X0W ) λ πe e n hy Y W ) µ πe e n hs Y W ) µ πe e n hs Y W ) We can easiy verify that 68) µ 0 µ 0 λ 0 σ ˆN 0 µ µ λ σ ˆN 69) Appying emma 9 to 67) we obtain λ 0 σ4ˆn 0 µ σz σ ˆN0 λ σ 4ˆN µ + σ ˆN 70)

15 5 From 69) and 70) we obtain ) λ 0 + λ σ σ σ X 0 Z Z λ + + µ σ σ σ Z N Z + 7) On the other hand we note that for each the five random variabes W X Y Y + and W+ form a Markov chain W X Y Y + W+ in this order Then appying emma 0 to 66) we obtain µ σ N e r + σ 4 Z + λ + 7) Combining 7) and 7) we obtain for ) λ + e r ) + λ + 73) σ σ σ σ Z N Z + Z + Set ν 0 λ 0 σ X 0 ν λ σ Z Then we have IX 0 W ) n og + σ X 0 ν 0 ) IY W Y ) n og + σ Z ν ) IY W Y ) n og + σ Z ν ) nr Note that ν 0 are nonnegative From 7) and 73) ν 0 satisfies the foowing recursion: ν σ Z e r ) 74) ν ν ν 0 ν +σ Z ν + e r σ N e r σ N + e r σ N 75) ν + +σ Z + ν + + e r σ N 76) ν +σ Z ν ν 0 e r 0 ξ σ X 0 77) From 74)-77) we obtain the upper bounds of IX 0 ; W ) and IY ; W Y ) in emma 7 On the other hand from 76) 77) and the nonnegative property of ν 0 we have ν 0 ν + e r 0 ξ σ X 0 + ν ν σ N e r ) + ν 0 σ Z ν 0 78) σ Z + ν σ N e r ) + 79) From 78) and 79) we obtain the ower bound of IY ; W Y ) in emma 7 C Proof of emma 3 et α β A α ) Then we have the foowing chain of inequaities: tζα ) + t)ζβ ) t og α ) + α + ǫα + t)og β ) + β + ǫβ t og ǫα ) + t)og ǫβ ) + +t og α ) + t)og β ) α og t + tα + ǫα β t) + t)β + ǫβ a) og ǫtα + t)β + + og tα t)β og b) + tα + t)β ǫtα + t)β +tα + + t)β + og ǫtα + t)β + og tα + t)β ζ tα + ) t)β Step a) foows from the strict concavity of the ogarithm a function Step b) foows from the strict concavity of ǫa for a > 0 Proof of emma 4 Proof of emma 4 part a): For the proof we use the foowing inequaity + a + ǫ + a) a + ǫa + ǫ 80) The recursion of 3) is equivaent to θ ω) ǫ θ ω) θ + θ + ω) + ǫ + θ + ω) + 8) for Appying 80) to the second term in the right members of 8) we have θ ω) ǫ θ ω) θ θ + ω) ω) + ǫ θ + ω) + ǫ + ǫ or equivaent to θ ω) + ǫ θ ω) θ θ + ω) ω) θ ω) ǫ θ + ω) + ǫ + ǫ 8)

16 6 for The vaidity of 4) is obvious From 4) we have ǫ θ > θ ω) ω) +ǫ 0 θ ω) 83) θ ω) < +θ ω) +ǫ +θ ω)) We first prove 5) Using 8) we have θ ω) ǫ θ ω) θ ω) θ ω) + θ ω) + ǫ + θ ω)) + a) < Step a) foows from the second inequaity of 83) Using 8) we have θ ω) ǫ θ ω) θ ω) θ ω) θ ω) + ǫ θ ω) + ǫ a) ǫ + ǫ + ǫ Step a) foows from the first inequaity of 83) Thus 5) is proved Next we prove 6) by induction with respect to From 5) we obtain and ǫ > θ ω) θ ω) < θ ω) +ǫ > 0 θ ω) +θ ω) +ǫ +θ ω)) 84) θ ω) ǫ θ ω) ǫ + ǫ 85) Soving 85) with respect to θ ω) we obtain θ ω) Using 8) we have ǫ + ǫ + ǫ ǫ θ 3 ω) ǫ 3 θ 3 ω) θ ω) θ ω) + θ ω) + ǫ + θ ω)) + a) < Step a) foows from the second inequaity of 84) Using 8) we have θ 3 ω) ǫ 3 θ 3 ω) θ ω) θ ω) θ ω) + ǫ θ ω) + ǫ a) ǫ + ǫ + ǫ Step a) foows from the first inequaity of 84) Thus 6) hods for We assume that 6) hods for some + with + that is ǫ + +ǫ + +ǫ + ǫ + θ + ω) θ ω) ǫ θ ω) ǫ + θ ω) ǫ θ ω) < θ +ω) < ǫ + From 86) we obtain ǫ > θ ω) θ +ω) +ǫ θ + ω) > 0 θ ω) < +θ +ω) +ǫ +θ + ω)) +ǫ + 86) 87) and θ ω) ǫ θ ω) ǫ + + ǫ + 88) Soving 88) with respect to θ ω) we obtain ǫ θ ω) + ǫ + ǫ ǫ + Using 8) we have θ ω) ǫ θ ω) θ ω) θ ω) + θ + ω) + ǫ + θ + ω)) + a) < Step a) foows from the second inequaity of 87) Using 8) we have θ ω) ǫ θ ω) θ ω) θ ω) θ + ω) + ǫ θ + ω) + ǫ a) ǫ + ǫ + ǫ Step a) foows from the first inequaity of 87) Thus 6) hods for competing the proof Proof of emma 4 part b): We first observe that ζα ) og ) α + α + + og ǫ α ) ǫ α + og α ) og ǫ α α + ǫ α )α + og α ) og + α + + ǫ + α + ) α Computing + og α ) α ζα ) we obtain α ζα ) α α ǫ α + α α ζα ) α α +α ǫ α + + +ǫ +α + α for From 89) when ζα ) 0 α must satisfy α α + ǫ α 0 +α + +ǫ +α + α α + ǫ α 0 for From 90) we obtain α α + ǫ α α ǫ )α + ǫ +ǫ +α + ) α + ǫ α ǫ )α + ǫ +ǫ +α + ) for 89) 90) 9)

17 7 The reation 9) impies that ζ α θ ω) 0 Proof of emma 4 part c): For the proof we use the foowing recursion: θ ω) ǫ θ ω) θ + θ ω) ω) + ǫ + θ + ω) + 9) Taking the derivative of both sides of 9) with respect to ω we obtain dθ ǫ θ ω)) dω dθ dω + ǫ θ + ω) + ) dθ + dω 93) Since θ ω) A θ ω)) we have θ ω) ǫ θ ω) < θ ω) The above inequaity is equivaent to + ǫ θ ω) + ) > From 93) and 94) we have For set ǫ θ ω) 94) + ǫ θ ω) + ) dθ dω dθ dω dθ + + ǫ θ + ω) + ) dω 95) Φ ω) dθ dω Then by 95) we have j +ǫ j θ jω)+) Φ ω) Φ ω) Φ + ω) for 96) From 96) we have Set Φ ω) Φ ω) Φ ω) Φ + ω) Φ ω) Φ ω) dθ dω +ǫ ω+) j +ǫ ω +ǫ ω+) +ǫ j θ jω)+) 97) +ǫ j θ jω)+) j Aω) +ǫ ω +ǫ ω+) Then by 97) we have j +ǫ j θ jω)+) Φ ω) Φ ω) + )Aω) ) +ǫ ω j +) +ǫ ω+) +ǫ θ jω)+) from which we obtain dθ dω ) +ǫ ω competing the proof j+ +) +ǫ ω+) +ǫ j θ jω)+) REFERENCES Sepian and J K Wof Noiseess coding of correated information sources IEEE Trans Inform Theory vo IT-9 pp Juy 973 A Wyner On source coding with side information at the decoder IEEE Trans Inform Theory vo IT- pp May R F Ahswede and J Körner Source coding with side information and a converse for degraded broadcast channes IEEE Trans Inform Theory vo IT- pp Nov J Körner and K Marton How to encode the modue-two sum of binary sources IEEE Trans Inform Theory vo IT-5 pp 9- Mar S I Gefand and M S Pinsker Source coding with incompete side information in Russian) Prob Pered Inform vo 5 no pp Y Oohama Gaussian mutitermina source coding IEEE Trans Inform Theory vo 43 pp 9-93 Nov Y Oohama The rate-distortion function for the quadratic Gaussian CEO probem IEEE Trans Inform Theory vo 44 pp May Y Oohama Rate-distortion theory for Gaussian mutitermina source coding systems with severa side informations at the decoder IEEE Trans Inform Theory vo 5 pp Juy A Pandya A Kansa G Pottie and M Srivastava ossy source coding of mutipe Gaussian sources: m-heper probem Proceedings of IEEE Information Theory Workshop San AntonioTX pp Oct Y Oohama Gaussian mutitermina source coding with severa side informations at the decoder Proceedings of IEEE Internationa Symposium on Information Theory Seatte USA Juy 9-4 pp Juy 006 S Tavidar P Viswanath and A B Wagner The Gaussian manyhep-one distributed source coding probem Proceedings of IEEE Information Theory Workshop pp Oct 006 T Berger Mutitermina source coding in the Information Theory Approach to Communications CISM Courses and ectures no 9) G ongo Ed Vienna and New York : Springer-Verag 978 pp S Y Tung Mutitermina source coding Ph dissertation Schoo of Eectrica Engineering Corne University Ithaca NY May H Viswanathan and T Berger The quadratic Gaussian CEO probem IEEE Trans Inform Theory vo 43 pp Sept H Yamamoto and K Itoh Source coding theory for mutitermina communication systems with a remote source Trans of the IECE of Japan vo E63 no0 pp Oct T J Fynn and R M Gray Encoding of correated observations IEEE Trans Inform Theory vo IT-33 pp Nov 987