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(2) 2 K. YosHIHARA DEFiNiTioN 3. We shall define the message (P, 'PV) as the aggregate which consists of two measurable spaces (X, S.) and (k Sr), a probability distribution P(.) on the o-field Sx and a subset 17V of the Set of''all the probability measures in the product a-field Sx x Sr. The spaces (X, Sx) and (iS;, Sr) will be called the spaces of input and output values of'the message, and the subset PV wili be called the condition of accuracy of reproduction.. DEFiNITIoN 4. Two random vafiables 6 and 6N generate the message (p, J>V) if they take values in the spaces (X, Sx) and (￡ SI) respectively, if hee pd5ilrib'beU18XOgns ?g Sv.COiPCideS with P(･), and if the joint distribution peE of. tth. In this paper, it is assumed that VV is defined with the help of MSxxS2mesurable functions pi(x, fi)(xE 1 X, Al fiE X) and by a set W in a manner comparable with that by which (1) defines V.. l, T 't. a ". i' )'1. ,,z' l". DEFi,N. iTigN 5. We shall say thag a message (p, VV) can be transmitteq Ijll)iti2hli,saa2igM,.lt"fhle.lg,o'.,iVo.w' /ii,g,t2gegr2,i,t?';Xi/Lt; .f.08r, r,ln,illOrl] ,()Ia,ila,bje2,Z',.Z' ij a"d8. (ii) 5hned Pair (8, g) Satisfies the conditign of accuracy of reproduction w, (iii) the pair -(ty, ij) is connected by the transmitter (Q, V).. ･･ Next, we shall define unique transmitting methods. Let e and T be sets of parameter valu,es. We assume that,with each rG l" there corresponds a transmitter (Qr, Vr)･ such that the input and output signal spaces do not de-. pend 'on r and that with each 0Ee there corresponds a message (P6, Wo) such that the ranges of message values at the input and the output do not. depend on O. ,. DEFINITIoN 6. We shall say that a family of pairs of random variables {(6e, g), 0E0} generates a system of messages {(Po, VVe), 0G0}, if the fol-. lowing three conditions are satisfied: , ,. (i) for each 0E e, (6e, 6) generates (Pe, Wo), (ii) among the (ps, vao), there do not exist messages with identical Pe. i,.butdifferentPVe,and - ,, (iii).the8earemutuallyindependent. .. I1DEFINITIoN 7. We shall say that there exists a unique method of trans-. ,,･. mitting,a system of messages {(Po, M, Ze), 0E @} by a system of transmitters {(Qr, Vr), rE Z-'}, if there exist a coding function P(.,.) and a decoding functionlvP(.,.) (independent of r and 0) such ･that for each rET and 0E 0, the. message (Pe, VVe) can be transmitted by (Qk Vr) with the help of the coding fgnCit.iO :aee(,'; 'lhaendsytshtee.dseCOodfiniliefsUsa"gC8isO"{ $;l 'w')e'), o E e} and' of transmitters t{ r(. gnrs' mVirt)terr G,, ire}sp)ggitiivO,efitye." be caii'ed as "pompoupd message " and " compound. Let C and cN be.random variables with ,values in the measurable spaces. ri,. 'i. l.>''. - '1･.
(3) The Shannon Problem for a Unique Transmission Method 3 (Z, Sz) and (ZN , Sz'v) respectively. By ,the. information, of one of t･hese variables. with respect to the other is meant the valite. ttttt/ tt tt t. (3) ' i(C,' c"") :,supi p. cc(EixFi) iog -;-C,tt-tt/l./t'tt,(FFI))'-. '. where the supremum is taken over all partitions {Ei} of Z and {Fj} of Z･. The following result is due to I.M. GELFAND, A.M. YAGLoM and A･. PEREZ: ' ･･ '. (i) If the distribution pcc is not absolutely continuous with respect to. the distribution PcxP& .theh I(4, 4ny) == oo･ .. (ii) If the distribution pc.f is absolutely continuous with respect to PcxP;-,. in symbol, Pctt<PcxPe, then ' ' ' (4) I(C, CN) == SS Iog a,c(2, 2N)P,c(dz, d2N). -･ ･ tN .･zxz ･. where ac,:(z, 2N) is the density of pce(.) with respectto 'Pc×P4(.). The function i;c(z, 2N.)== log acr(x, 2'") is called theinformation density'of the random variables. C and g-. acc(4, 4) and icc(C, 4) are measurable functions of the random vari-. ables (C, 4N) and are therefore themselves-random variables, which we will denote by a,･;(C, 4'") =a(C, 4N) and i,i-(C, cN) ==i(4, 4N) dropping subscripts. We remark ,here. that I((;, 4"") l.llo and if I(C;, CN) < oo, then i-4e(2, 2N) exists.. Let '. ' '.',, ･ tt ･･･ lt tt '. '. (5) C(Qr, Vr)=SUP I(rp, rpNr). where the upper bound is taken over all the pairs of variables (rp, ijr) connected by the transmitter (Qr, Vr)･ By the capacity of the system {(Qr, Vr);rE Z"}, we shall mean. tt (6) ' 'C(T)==supinfl(rp,ijr) ' ' '' ' 'rp rer where (rp, rp'"r) are connected by (Qr, Vr) and where the upper bound is taken over all the variables rp for which there exists such a pair (rp, ijr), for all r.. Fgrther, let. (7) H(Pe, We) == inf l(6e, 6N) `,. ' lower bound is taken over all pairs (ge, 6N where the ) which form the message # (P,g, M7r). By the entropy of the system {(pe, I7Vo), 0E0}, we shall mean ,,. (8) H(e)=s,u.,pH(Pe,M7e). "''. .-･" t ･,'･. ttt . t gttttt ,h.iktfhO,r'.e.abChvioO.igiye ahd r E i", rahdom vafiabies 6B, op, op"",, f6fni`a Ma,kov. (9) .H((E));$C(T) ･i. '. holds.･'.'･･ ''''･'.'',,''･,'/",･･･t'..'. " - Next, we shalli consider a sequence of pairs of ' random vatiables (Ct, <N't)..
(4) 4 - ･K.YosHIHARA Ag in [3], we shall write Ptc.v,it4e(2, 2N), I`(C, 4"') ･･･ instead of P4tEt,ictet(z`, 2"t), I((;t, 4--t), ..... DEFINITIoN 8. The sequence of pairs of random variables (4`, 4""`) (t =1, 2, ･･･) will be called information stable if O< I`(4, C) < oo and. (lo) .'･･･. ,･ ,li..co.it,((cd,/2rf)=1 '. in the sense of convergence in probability. DEFINITIoN 9. It will be called information stable if there exists an information stable sequence of pairs (ty`, ij`), connected by the transmitters ti. -(Q`, V`) such that C`(Q, V)< oo and. (ii) .1ii.ee c{i(Q",' ?-)7-i･. t'. a. '. DEFiNiTioN 10. The sequence of messages (Pt, Wt) with H`<p, VV)< oo will be called information stable if there exists an information stable sequence of pairs (6t, 8N`> forming the message (Pt, IV`) such that (i2) ']. '''. ,. ., ligtiH4((fip6'fi/-r=i･. Fipally, tbe.following definitio'n is'intr6duced. Let UdRn, where Rn is the ' n' -dimensional. Euclidean space with points x-=(xi, ･･･,x.), and define the. distance between the points x-i== (x{, ･･･,xA) and Xi' == (xl', ･･･,xai) by. ' nf/i) = max1x(･-x:･'I･ tt d(x'rii,. For any e>-- O, let ' '. i. [U],=={x-ER":d(nf,x-')ge forsome x'"iEU}, We denote by (Q, V,) a transmitter for which in definition (2), the set V is replaced by [Y],.; by (P, VV,) we denote a message for which the set VV is replaced by [W],... ;,. '. ' ttt tt 3. The direct. '. Shanon theorem for a unique transmission theorem.. The object of this paper is tb prove the following direct Shannon theorem. forauniquetransmissionmethod. , , THEoREM 1. Let (E) and T be two arbitrary .fZnite sets. PVe assume that there exist a sequence of comPound messages {(Pte, VVS),0G 0} and a sequence of comPound transmitters {CQtr, ･Vi r E Z-'} with the following Properties:. ,I. For all t, Ht(e)<oo and Ct(T)<oo. Moreover, limHt(e)=oo and lim Ct(r) = oo. t-oo. ii. LiLm. g,ti,e.i <i.. ' t'-> co. III. For each 0E@ and rE Z"', let Mte and Ng be the nzambers of functions .phk(xe,fi) ,and of functions ziic(y,yNr). Then for any a>O the relations Mt. --. k.
(5) The Shannon Problem for a Unique Transmission Method 5. =max Mb==O(2aUa@)) and Nt=max Ne ==O(2aCt(r)) hold. .. OEIOV. For each rET, there re`L-xrists an inj'ormation stable sequence of pairs. of random variables {(rpt, ij9)} (each (rpe, ijtr) being connected by the transmitter (Q9, V;)> zvhich satis.fies. LiLm.. i5/7tf/Zrv)r' >=i. and for some b">O and c-`==O(2"a`(r') (where a>O is arbitrary). ' max max E{ln9ic(rp, ij,)-Erci,(rp, rpNr)ii"'b}S c-t. t rEr lS-leS-AIr ･ V. For each eEe and t, iVh is a convex set oftheMS-dimensional Euclidean sPace. For each 0E e, the sequence of messages {(PS, VVS)} is injormation stable and there exists an injeormation stable sequence of Pairs of random variables (6b,6`) which satisfies the condition of accuracy of reProduction (PS, M7S) and for some b>O and c`==O(2""`(e)) (where a>O is arbitrary) (i3) r}l.a,X,.M.,l= ib .r[I.!,Pbic(Xe' xN)-.S, !,ebic(xe, Je)P}eE(dxe, dfi)1i"bpge6(dxe,dxf)i c` .. ' (i4) ilil3,X,..M,,a..X,, .S.Sl,Phic(Xe' fi)-.S,fsbic(xo･ fi)PS×Pg'(dxe･ dN)li"bps×pg(dxe,dfi);sct. '. (15) iil}.a,X,.M.,-.a-X.,,.S.S.!,Phic(Xe'N)m.S,fShk(Xfi,I)PS×Pg(dxe,dl)iPg,g(dxe,dil);$ct ''. '. (i6) iiP3,X,..M,..a.Xb .S,f2,Petic(Xe･ fi)-.S,f.p.ytic(xe･ xN)PSxpg-(dxe, dx)l`"bpb'×pg(dx,, dfi);$ct. VI. ForeachOGOandt, , ',.. ' ''' .,' . '. (SS Pbi(Xo, fi)PSXPE`(dxo, dxN), "', SS pteM,(xe, xN)PSxPg(dxe, dfi)) G fi7S.. '. xtxxt irtxxt. VII. For each 0e0 and rE I", the above definledrandoin VahabZ2s (rp`, ijei. and(gS,eN`)satisflrvtherelation ･ ' '. ' i[rprpl,oprp"'"-}))m-',,'.6).1==9' ' >6).+pt?Nr( ?Lm.. [.(c,`)a+(Mt)"]'[p}eg( Slfil fN'3s7-i. wheree>O,a>OandS>Oare.arbitrary, ･･. .,.,.,. Then, for any e> O, we can find a number to so large that for all tl.ll t, we. Cmalltetrra{nieMi℃tJ4,,e),CrOMEPilZ¥.nd. MeSSage {(PS, VVSe), 0 E e} over the compound trans-. ' ""' t' 4. Feinstein's lemm,a for a finite system of transmltters {(Qi., V;),xET}･ Let T be a set with d elements. Let {(rp, ijr),･rET} bea family of,pair$.
(6) 6' ･i K. YosHmARA of random variables conriected by a system of transmitters {(Qr, V7), r E T}. ,L .,gt.ij.*,.2e,.a. ,r,a,"di911],,V.a,/ia8,i2,gege.S,d,gp tP,eJ;EoBa.?iiity space (s?, EBi p), yaking,. (?*(' ' ') i= -d-,?TQr({ ' '). '. '1 .'･. '. P"7'(')= 2 ,?rPh7'(')'. l t/ tton the measurable space (yx 9, then pfi,(.) is a probabl'iity 1' . distribution. S.xSfi), '', 1 .. '･i''. ". , /i,,;, PT'",,(･)=d,iilljPffT(.')S ". andf granyAGSr' , p{ij..Aiv..y}=Q9(y,/i)･. tt.. In [6], MA Xi-wEN proved the following lemma. LEMMA (MA XI-wEN). 11/, for each r Ei T,in7,(.,.) exists, th2n Pvicw.<P,xP,"",.. ifwedenotetheinj'ormationdensityas ･ ･. z' ":' 'i*('･')= dpd,eeS(i,i:) ' numbers ai and a, then we obtain that for two arbitrary Positive ' PTif*{i*(rp,rpN*)<ai}S2,?rPTyr{i(rp,rpNr)<ai+a2}+d2-"2.. ' To prove TheQrem･1, wei need,the following FEiNsTEiN's lemma for a finite system of transmitters which is a modification of MA Xi-wEN's result.. THEoREM 2 (FEINsTEIN's lemma for a finite system of transmitters). Let {(Q;, Ve), r ff I"} be a finite system of d transmitters. For any S> O, Put , .･. Lt6 = [2(t-6)at(r)] .. tt. 4. et {Pl, ･･･,Pls} be an arbitrary system of non-negative numbers such that. i. ' 2L3 ., ,. a7? , , , ･/tL/..=X.,,?{SL3;tl-iPf=l,'i: . ,:.,-{ , ' 'for any a>O, M`=O(2aC`(P). FinFurthermone, let M` be a number such that gyi?t>m'Ol[f 1"o%-nmat:'ivele ;iu"m`Beri8tszell.(hPSIh(E;fL'r''ah?'i,=I･}I L'1,"" :i'i" be 1. (i8) ,(i,,ej),tog"･<=2Lig6-C`(Q･v)' ". .' .I, ,O. bh'. d/1,h,e,h,i,"?/i,e,r'ft,h,e ,anvy-.p,t,io,z2,i" y.,of,,,T.he,o,r,e,7.i･ eq.e can find a iarge num-. l i /. '. j :. t 1.
(7) The Shannon Problem for a Uniqqe Transmission Method 7' {(YSAf),'",(Yt.t,,Alt,)}(YIGYt,ASEStg;i-ri1,･･･,L3) g,/, which satisfies the following conditions:. (i) For fixed t, At･AAS･=ip (i #';i=1, ･･･,L3; ]'=1, ･･･,L3).. (ii) For each tll t, and rEI" .. t:-i L3. (19) zpfQi(yi, Ai) lil-S･ i=1. <iii) Fo}eacht>=t,andrET '. (2o) (li.l6,pzS,,nli(yt･y'-)(?i(yz･dyN)･ ･tll.i,psS,,n;Nr(y,･y'-)Q)i(y,･dy))ci[i7e]6. (iv) For each tli t,, rGT andi (1:.{i.S L3). ' (21) ' Qe(Yi,Ai)lll-,li.il.PSvA'r(t'i'l'&rpirp?-,r]--1>i6T)-2-SooLC`(T'. '. (v) For each rEI-', le (1Sfe$M`) andtlt,,. t tt/t. ' '6' ' L3 L3 .t. (22) , ,l=,j=,icpg･Re(yi,/lj)i:;ll3･. (i# j) '. ' ' PRooF. Since, by Assumption L Ct(T) < ool so It(rp, ijr) < oo, for all rG I7t. Thus there exist the information densities itopnN,(.,.), r E T. Let. (23) ,. F`-{(y`,Yt);i.t(y,yN)IC"(I")(IT:.)}. Then, we can easily show that F`ES`y × SS and '. tt tt. t t ttt .t tt.t. (24) ' pt,(.El);:$2dO`(T)(i-e)' ';･. tt t tt for almost all yNtG Y` and I7e-ES`v where FS- den'otes the Y`-section of the 'set. Ft. ,, . ., .. ' on the probability ' variables, 'defined Next, let 4f, ･･･,Ct.t, be L`6 random. space (9t, 8t, B`), which take values in the measurable space (Y`, Sf) and have the same distribution' PS(.)7 Let. z,(toNt,y-'`)=(4f･(di),yNt)' (i=1,･･･,L3). ' In [3], DoBRusHiN proved that the Zg･ are measurable and if, for any g` E! E8(･×Se-, (25) SPg･(g) = Sy, Qk(4,(caN), G(g'"))p"'`(ddi) =EQk(c,(di), .t G(tuN)). where G(w'"t) is the to""`-section of g`, then we have that for any CtGSyt × SS. (26) . PS,-.(C)==pm:.(Z:.i(C)) ' .'T/ .. :.t. '' 'ttl/ tt andthatforanyDEEBtxSS -- -, /t ..
(8) l. 8 '' K.YosHmARA. l. (27) '''''EBg(D)=S.as,--,(4,(di),y)p'"txp,.tv,(d.-v,dy). '. '. where a"t,,--, (.,.) is the density P9,A',(.) with respect to Pv` ×P9, (.).,. Now, if we put . &g.=.Z,:i(Ft> (im-1,･･･,LS) and. '. (28) M&t,=&g+-U&S･ (i-1,･･･,LS), j' iki.. then, for each i,&2･EE8t×SS and thus &e･EE8`×SS. Furthermore, for eachi. `. the di`-section of &t･ coincides with F&(x) and the tu'"`-section Ft･(di) of the set &g. satisfies the relation. (29) F2･(('v") =FE,(x)-UFk,,'.v,.. a. j' 7Ei Since the Cg are independent and have the same distribution, so by (25), and (28),. (30) asg<&)lasg･(&)-Z8t･(&A&)).P9,r-.(F)-L3asS(&A&,)･. j'¥i We shall evaluate asf(&A&,). We remark first that (di`,yN`)E&fA&S if and only if <Cl(toN), Y`) G Ft and (CS(toN), yN`) G Ft, simultaneously, and thus the yN`-section. of the set &fA&S coincides with the set ･ ･, . ' {(4i(w'")GF8)A(4S(tu'V)EFt-)}.. '. As, in general, we have that for any nonnegative Sf-measurable function. u(y) and for any BG SY× SS. .t .. S{(,,,,,)..}Z{((IIf(di))P'V`(d('vV) = S.U( YS)lbS × PS(dy,, dJy,) == S.,za( yS)PS(4,)pS(dy,)',i.. so putting B=17&- ×Fe- anq u(yl) == a"tv,N,(y, Y), and using (24) we have. -'. s. S,,l,,s,iS"N*(4i(tuN)'YN)P'V`(dW'")=S.,a"optn'v.(yi,yN)PS(Fg)pS(dy,) ,,... , ;$2-ct(r)(i-e)S.,as,-･,(y,,y'v)ps(dy,), '''' ', '. for almost all Yt. Consequently, by (27), we have. , (31) 'El3f({5iA{52)=S,i.,sa"Tt?'v.(C,(to'"),y'")P'"`×pt,'v,(dwA",dy'"). ' =Sg,PtVN*(dY'V)S[g{,gs,ar,aS7,(Ci(toN),y'V)P'"'`(dto"") .. ' '. :"f{; 2-C`(T)(!ML3fi) S?,p`n'v. (dy'V)S.,a"ntoj.(y3, yN)Ptn(dYi) = 2-ct(T)(i-t' ) ss' pt,N,,(dy,, dJy) ..2-ct(r)(i-e-) ,. Ytxyt. i.
(9) The Shannon Problem for a Unique Transmission Method 9 Combining (25), (30) and (31), we have (32) E([?sc(C,(w'"), F,((b)) = EP,([ii,) ). pbl.(F)-Lt,.2-C"("(iM-SL) l.llr ps7,(.F)-2--62-Ct(J').. 5P,¥･,,ilmg,Eg"5/r3"b,9.`(S3t/f,.{`(rpi:-rMiOk,9.i',<,8.,C,h.:hk2･ Ag.ge PUt ai=. (33) Ptop'nv,(F)li1--li--)g.p`,',v,(z(rp,ij,)<ct(I-')(1-'.g))-d2-"Og`". lli1-Lt,lii).PtnAv'T( Zi((Z', ijrp.r,)) -1 > -g-)-d2- 6C`,'￠-'. Combining (32) and (33), we obtain , (34) E(?k(Cz(toN), i7z(di)) ;S1-{l-,?.PtTA?",( !i<-(rpi ijrp.r,)) -1 >-g.T)-2--2B(ob7Ct(r'. and ''. (35) E{ tli.li pg(?g(4z(("t"))･ Fi(('ti"))};-:{i-tt-,E.) ps'v",( tt-1.1rpe, Z.i)) -i >-g-)-2-- 2go Ot(r'. for sufficiently large t. From (34) and (35), we have (36) P'"`{Qr(Cz(to'"), "i5z(tuN)) l.ll: 1-,]i.1,.Ptv'nv,( {<Zrp', Z.r,))--1 > -66')m2-To"ofC`"'). ･ ,, forallandrEI"}>=19o. '. (37) p'Vt{,S.lip,Q,(c,(wN),fi,(di))>=i-sforanrEi-'}i-iiiri9o-. On the other hand, by (24)'and the fact that, if i 7Ej, Cf,(wN) and CS･(wN) are. mutually independent, we have ･ ' '. '. . ' EQk(4,(toN), Fj(tuN)) i:;I EQt.(C,(to""), Fc,t(ar)) '. '. =SSQt.(y,F,,)Pt,xPt,(dy,,dy,) ] , Ytx fit. ' ==S,,pt,r-(F,)pt,(dy)ls2-ct(r)(i-g) (i#j).. Accordingly,foreachk(ISfeSMt)wehave , . E{ ,iSi Itl¥'i, icpg,･Qt.(k(di), .F,(.･-))} :$ Ls 2iii6-ct(r) .2-(i--g-)ot(i-o. (#j'). '''' '' ' ''iK2--2･(]6-oct(r)-,. whichimplies., '･-,',/'･ .. -----.
(10) K. YosHmARA,. 10. (38) P""`{,l.l,i,,ISIkpS,Qi(4,(to"v),L(di));s-g-foraiireT}l.iiii9o. for$uMcientlylarget. , In [3], DoBRusHiN proved that if we define Rrj(.7f)=S,,njj(y,y'-)Q`r(y,dyN) (i=1,･･･,Ari;,rEI")'. t/ ttt forallN`EY`,and '" ' /: '' t ' '･''1･, '. i. 6S･(di)=2r,･<Cg<di>> <i=1,･･･,L3;7'--1,･･･,Nl;rEr). then for any 1' (1gl' .SNe) S' ,･(di) are randoni variables for which. v. ,it. ESy't,･(wN) = Ezh(rp, ijr). and ･'' ,. .a. (3g) p'V`{ ,j$.ii p,s&,(tuN)-Ez;,(rp, rp"'r) > 5} ;:S (LD,,()C-i;.(?,-,. hold,where D is a constant which depends on b- and 6 only. Sinee. ' -.. (Enii(rp, ijr), ''', ETiN,(rp, ijr)) E [ t7e]. so, using (39), we have P'"`{(,IL=6,p,Szi,(cNo),･･･,,L.ii.:,piSi.,(aJ))d['i7,]a for allr(ET}l.li 19o ''. that is,. '. (40) p'"`{(,z,=6,piS,,zV,(Ci@N),y'")Qr(4i(w'v),dy'-),. ''', ￡i PiS,,ZrNr(Ci(toN), Y) Qr,(Ci(tuN), dyN)) e ['iC7i･]6. ' for all rGI"}l.l) 19o. t. ;. for all suMciently large t.. Consequently, from (36),,,(37), (38) and (40), we deduce that for all suthciently large t, there exists a lvNo` G O` for which the following four inequalities .. hold simultaneously:. L3 ' P'. (i) 2Pg･QE(C,('toN,)) Fi(wN,)) >=IT6 for all rEZ-', ' i=1 (ii). Q9 (Ci(w'"o), "l7i(toNo)) lil 1' ,li. III. .P 5Yr(. I(ty', rp"'r). ' ---2-'`iet6-oct(r). (iii). i(rp, ijr). -i .g) 1. for･'al1. L3 L3 -. Z Z ic'pS･R;(Ci(w'",), F,(w'",)) ::$. 7GL. -3S- for all. rE r. i= (:.ijil5i. and for. all. le (1SlesMt),..
(11) The Shannon Problem for a Unique Transmission Method 11 (iV) (,￡iS･S,,Zi,(4,(toN,),y")Qi(4,(tu'V,),dy-),.... ' YN)'(?7.(.4Z('10)1 blN2,E.i,i'It],,', ... '"' ,IZ'i PtS?tz'tr"r(Ct(toNo)'. Now, putting yg･== cg･(w'v,), A:. =Fg.(tu'v,). we have the desired system. So the proof is completed. ,, '. '. 5. A theorem concerning a finite system of messages {(PS, WS),0e 0} In this section, we shall prove some lemmas and a theorem concerning a sequence of finite systems of messages {(PS, WS),0 E 0}. For each 0 G e, let (ee, e) satisfy the condition of accuracy of reproduction (Pe, PVe). Let e ={1, ･･･,h} and {aj} be an aggregate of nonnegative numbers. such that hZad=1. Further, let. jLml ''. (41) gd(･)=Pi×'"×Pd-i×Pj'+i×'"XPh×Pejs(･)(7'---1,'",h). and .. .. .:.. . tt t (42) P*(.) == 2a,･g,･(･)･ h'. j'---1. Then, obviously, p.(.) and g,(.) (]'=1, ･･･,h) are probability distribution on. the .(h+1)-dimensional Cartesian product measurable space (Xx ･･･ xXxk. Sx×･･･×SxxSi) and for any AGSxx･･･×Sx ,,･ P.(A × g) =P, × ･･･ × P,(A) = P,,,,",,,(A). WhenP*<P(i,m,h)xPe,weshalldenote ..,.. ..d , a*('. dP.(.) I')=dPqs;,h)×,P.f'(･) ... ' i*((xi, ･･.,xh), fi)=log a*((xi, ･･･,xh), it) , LEMMA 1. 111C, for each 7' E e, iejs (.,.) exists, then P*<Pq..,h) × Pe and for. any a,>O and a,>O ' (43) p*(i*((6,,･･･,6h),eN)2'a,)s.,Sj,'a,p6k'(i(e,eN)la,L'a,)+2'a2. holds. ･ ･ ., .. ,,<PpR,,9;8,'.Ii)"i. tt. ;'8S (sfil･ El71'･eEi 0.k,eXS:g',S.O.P,egiS,9kX,9/i' f,o,r, ai' jE`E)･ Th"s. and D={((Xi' '"'Xh)' fi):i*((Xv '",Xh), fi) > ai}.
(12) 12 K. YosHmARA Dj･ = {((xi, ･･･ , xh), fi) : I',･((xi, ･･･ , xh), fi) < ai- a,} A D. where i'j(('''"'')'')=dp.,..gf,jx('},i(.) , .. /t Since D and Dj(7' == 1, ･･･,h) are the sets in the (h+1)-dimensional Cartesiian product of 'a-algebra S. × ･･･ ×'Sz × SI and P*(D,･) ll 2"iP(,,...,h) × P,a (D,･) lll 2"2g,･(D,･) ,. so gj(Dj) ;$2-"2. Now we use the following fact proved in [3]. Let Px and x be any two probability measures on the space (X, Sx). Let Py and pdy be any probability measures on the space (Y, Sv). If. P-. ' dP. × Py(･) axy(X,Y)= dp-.×P-r(･). .. exists, then , dPx(･). dP y(･) az(x) = dp-.(.) , ay(-Y)= dp-.(.). exist and for almost all i(x, y) with respect to P-x × p-y. axy(x, y) = ak(x) ･ay(y) .. we have ,. Putting Px(･) = P-z(.) = Pa,.",,--i,j･+i,.",h)(･), Py(･) = Pe,je- (･) and P-y(･)=Pj･ ×P6(.),. '. ij･((Xi,･･･,Xi･,･･･,Xh),fi)=i6i･6'(Xd,fi) ,. for almos･t all points x-=(xi, ･･･,xh) whose 1'-th component is xj. Thus, we g,･(D) :S g,･(D,･) +g,･(i,･((6i, ･･･, 6h), gN) > ai-- a,) l. ;:$ 2-a2+pe,･e- (i(6,, g) > ai-a2). and consequently. '' h. '. h p.(D) == z ajgj(D) :;l 2-"2+ Z ojPej.fi･ (i(&, 6'") > ai-a2) J'-1. j'=1. ' Fh,is,5h,eh7,eSoir.edejn.e.qd"akiit(Yi'gfesMs),i,l ' '. whic. tt t. '. E*pbic(go, 6N) == S ･･･ SS pSk(xe, fi)P`*(dxi, ･･･, dxh, dfi) .. 'zt×･･･×xt×xt. t.. LEMMA2, UnderAssuznPtionsVandVI,wehave ,. . ''. /' ' 't tt. ' 6N), ･:･, E*pSM,(6e, gN)) e iZ7S (E*pS,(ee,. foreachOEe. ･., ,, PRooF. Since for each le (1:$k;llMS). ".
(13) The Shannon Problem for a Unique Transmission Method 13 E*pbic(6e, 6N)== aeS.i' toSk(xe, fi)P`6eT(dxe, dje). Itxlt + (1 ff Oe)SS PSk(Xe, xN)P te × Pte- (dxe, dxN) ,. .2rtx.2't. so, by Assumptions V and VI, we have the desired result.. LEMMA 3. Under AssumPtion V, we have ' S '" SSlpbk(Xe, fi) -E*pteic(6e, 6N)li"bPt*(dx,, ･･･ , dxh, dfi) :!$ ct. xtx･･･× rtx.2't ･ '. ' S[S ''' Slpbk(xe, fi) - E*pSic(go, 6N)1pE,,".,h,(dx,, ･･･, dx,)]'"bpg(dfi) ;$ ,t. .tttxtx-xlt ' ' '. PRooF. By MINKowsKI's inequality, '. {S ''' SSlphic(xe, fi)-E*p3k(6e, 6N)li"bP*(dx,, ･･･, dx,, dfi)}.itb xtx･･-xltx.2t. g- ae{S "' SSIptoic(xo, xN)-SSptoic(xe, fi)Pteoe(dxe, dxN)li"bpt*(dx, ･･･, dx,, dN)} ilb. xtx･･･×.xtxx"'t xtx2t , .. +(1-Oe){S'"SSIp3ic(xe,N)mSSphic(xe,fi)Pte×Pg`(dxe,dfi)li"bP*`(dx,,･･･,dxh,dfi)}iib. xtx･･･xltx.2t ztx.2tt =<= afi{aeSS1phk(xo, xN) -SS pSk(xe, N)P},{(dxo, dfi)1i"bPt6,E(dxo, dfi). Ztxlt .ItxXt +(1-ae)SSlpSic(xe, x'") -SSphk(xe, fi)Pte,e-(dxo, dfi)l'"bPS × Pgt(dxe, dfi)} !tb. Itx?t XtX.iit . .･ +(1-oo){OoSS1pteic(xe, fi) -SSptek(xe, N)PS × Pg'(dxe, dfi)li"bP`6,any(dxo, dx"). .rtx.2t xtxi?Lt. '. ' dfi)} i+'b +(1-aifS1phic(xe, fi)-SSpS,(xs, xN)pS × pg(dx?, dx)Ii-i-bps × pg(dx,,, '. ztxXft xtx.Xit '. 11. ,<.,, ag{oec`+(1-aoc`} i"b +(1-oe){aec`+(1-oe)c`} i+b. i . = (ct) 1+b Thus we have the first inequality.. By the analogous method, we have the second., ･. - Under AsszamPtionsJV and YIJ of Theorem 1, we deduce that LEMMA 4. for any 6 > O there exist seqzaences of sets B3 in the (h+1)Ldimensional Cartesian Product of a-algebras sl × ･･･ × Sk × S.e and constants ts such that (i) for (xS ･･･,xk, N`) E B3 and all le=1, ･･･,MS(0 E e).
(14) 14, K. YosmHARA lpbk(xe, fi) - E*pSic(6o, 6)1 ;:il JS,. (ii) forallrE1" l!Il2 JS[PopYr( iS[U,i Z-X,))77;7i > S) +2-6"t(e)] -=o,. and (iii) for any a>O. ' ' lim[1-pk(B3)][(bt)'a+(Mt)a+1]..o. teoo. PRooF.･Let ','･･i/ ･ ･･･.･,{ '. J$= Mi" ([ ip.,,,t,.xPbl,( t-Sl'rpny', t,Nr,) -i > s)]-rS", 2iga'"`(o)). n. andforanyO･Ee;./, .'.'/ ' ,r. ,. -. Bte6 == {(xl, ･･･, xE, N`) :lphk(x.A, xN)-E*pSic(g,, 8)1SfS; le == 1, ･･･, MS} ･. Moreover, let B3 == V Bh6･. rE lilY COnStrUCtiOn, (i) i9 SatiSfied･ Since A$sumptionIV implies that for all. ' ,, ,,ILIilll[PSAn'r(il',7irpS-i))-i>s)]'iS-=-o.. . so (ii) follows from the above relation and the definition of fS. Finally, we. remarkthqtforanyOEe,,,, ,, / 1-pt.(B6),<,,.1-Pt*(Be6) ･' '''. ± oePk6e(lpbk(go, 8N)-E*pSk(6o, e'")l>IS, le == 1, ･･･,,MS) '･ il'. '. '. +(1-ao)pS × pg(lpteic(ge, 4)-E*pSic(6, 4)I<JS, fe = 1, ･･･, MS). tt t･ . ,/ kctM-i'2 c`M` == (fS)i÷b =: (JS)i+b･ .. '. ;. Thus, by the definition of IS and Assumption VII, for all rE I" we have .L ･ 1-P*"(B6) :iilc`M`[PtopNop,( i((rprpi rpS.r,)),-1 -6)]-"±2"b'+ctMt2-iliotbL"t(e)-o J. / as t->oo. Hence, by Assumption IV, we deduce that the c- ondition. satisfied. ,,, /t,･ , fi`+ in .SXt (indePendent of 0) for which i,d. (iii) is. LEMMA 5. ?ilC AsszamPtions llL V and 'trII hold, then there exists a Point. liLg:[2-""t(e'+pko,( ii'(6S", ?l--i > s)+ph,(-il-rp,7', Z.i)) -i > s)](vt)d-o ''. ' '' holdsforany0EO,rGT,S?O.anda>O,where ' ',. V`=:il9eX,tl,2..Xe,S.tlP`ek(Xe,fi+)HE*pbk(6e,6N)ii"bpS(dxo) / ,,, '･'.. '. I.
(15) TheShannonProblem'for･a･UniqueTransmissionMethod ll5. Furthermore,foranya>O , ･ lim [1 ---p.t(B6)](vt)a= O.. tHinequality, OO , for each 0Ee and PRooF. By Lemma 3 and TdnEBYsHEv's. le(lskgMs) . .,･ .,,,.- -..･･ ･.-, i, ' Pg(S.,lpole(xe,6N)-E*peic(e,e.,6N)1i'bpe(dK,e))-2hcM)$2hlMt ,.. '. holds. Thus, there exists a point xNt. such that. . V.a,X,n.,'a-.-X.,tS.tlPtek(XO,fi+)"E*pteic(6e,6N)1'-b･ps<dx,)$2hctMt. Since v`$2hc`M`, so, by Assumptions III and IV arid Lemma'3, we' have the. desiredrelations. ･ ,,･ ･'･,. THEoREM 3. Let (E) be a ./7nite set, say, (E)=={1,･･･,h}. Let. (44) 'FS,={((xS･:r,xt,･),:i`.((x,,･･･;x,),fi)5(1+25)Ht(e)} i '. and. (45) rbt<x,,･･･,x,)=･･P`{((e.･･･,6h),gN)GFsl(6"･･･,6h)=(x"･･･,xh)}･ .tt/7dgms.SY,Ma"n'EOZii S' .i6il ,ihU}￡,I,i,ids.I,L,Y,f, Ca? 'fi"d"""mbfr fo s"ch tha'. Points fif,'. 1･･,fik3 in SX't, and for each fi$ there corresPonds a measz{rable func-. tion qg(xi, ･･･,xh)((x{, ･･･;xk) E Xt× ･･･ × Xt) for which the following conditions are satisfied.. ,(i)Foralmostall(x5･･･ ,,xrk)anaforallt,l.ll)te ･ '. (46) O;.:{q2･(x,,･･･,xh)rm<.1 (i-=･1,･･･,KS) ' L (47) Qt(x,,･･･,xh)=llli3iqg･(x,,･･･,xh)+rs(･xi,･･l,xh)s-i･ i==1. (ii) Forqfixedrc>Oandalltl-ilto･. ' ･ '. (48) 'Q)t-=S･･･SQ)t(x,,･･･,x,),p?,,...,,,(dx,,･･･,dx,)).1-:-2-dit(e). Ztx･-xXt,. an(d4giOranYa>O･L. ･lifr1[1-Q,]tv,).=･o)･''`'. t-l 'oo t. (iii) For all t). t,, 0E e,i=1, ･･･,KS;x`E Xt and 'le, ==4 1, ･･･,MS,.
(16) 16 , ･K.YosmHARA ' (50) S･･･Sltobk(xe,fii,)-E*tohic(6e,6N)lqE(xi,･･･,xh)Pk,-･,h)(dxi,･･･,dxh) XtX･･･×It. ' SJts.it,S ･･･ S qg(xi, ･･･, x,)PC,, ,,,(dx,, ･･･, dx,) .. xt×--× rt (iv) For all t>= t,, 0E e, k-1, ･･･,MS and all ifor which q"-,q>O (51) S･･'Slpbk(xe,fii)-.El*pbic(6e,6N)lph,...,,,(dx,,･･･,dx,)m<-2ToBo-H`(,e) Xt×･･.×Xt. t tt. t tt. '. qd`i=S･･･SqE･(xi,･･･,xh)Pti,".,h)(dx,,･･･,dxh)>O.'. Itx."xXt -.. ' (v)Forallt>=t,andOEO, . .. .. (SSp ''',ShMe) E [WS]a. where . K3. (52) SSic=,;.,Si･･･SpSic(xe,fiz)qg･(xi,･･･,xh)P?i,･..,h)(dx,,･･･,dx,) .r't×･･･×xt. +S'''SS pke(xe,fii)qS(xp'･･,xh)PEi,･･･,h)(dxp･･･,dxh). xtx･･･xxtx.2t-F3 ' +[1-Q`]E*pSic(8o,6) '(le=1,･･･,MS)･ .. '. PRooF. We remark first that the set F6t is an element of the (h+1)dimensional Cartesian product of a-algebras Sf× ･･･ ×Sk×Sf and for any :Ni,ff..XNg.'. g,f,.W.9 gg",9,t8,g.he,%,SZC,tiO,n, O;-.F,g8,y,.F,`i,･;h.en..tF9x,ig,.an eiement of the. tV and a As before, for each t, we introduce a probability spaces (9`, A. 8`,'P`) collection of KS independ,ent, random variables Cf(to'"),･･･,4k6(w'V) which are ..'8`,tvP`), have the same distribution pg(.) defined on the probability space -" (9t,. t. andtaketheirvaluesonthespace(gt,St.-). ' Now, for any 0E0 and le (1$leSMS), we put Db, == {((xC ･･･, xk), xN t) : pbic(xe, xN) - E*phic(6fi, 6)1> PO. and Uhic = {((x{, ''', xrk), fi t): S.,lpSic(xe, fi)- E*pSle(ee, 4)IpS(dxo) > 8t}. where 8` is an arbitrary positive number. By Lemma'3 and TcHEBysHEv's inequality we have (53) P9(Deic)-<-(pl]i+b.S,.'.J.L.S,Sl.fhic(f,e･xN),-E*pbic(Se･6N)1i"bPk(dxi,･･･,dxh,4m '. < l '' ･= (t(31)i.b. ct '.. -.
(17) ,. The Shannon Problem for a Unique Transmission Method. I7. (54) SIIj,SSIpS,(xe,N)-E.pbk(ge,,4)IPkSdx,l.t･,d{.ig,dxN):g(kC,,ii'. and tt ' (55) 'P`*(Uok)$(stli+bS.-,[S.,lpbic(xe,1)-E*pSic(6e,4)l i. .'' xps(dx,)]i"bpg(dfi);:s(sf)`ubt. Weuseheretheset･Bb"introducedinLemma2andput ' G`±,Y,{(DSi V i`' V DgMe) V(USi V ''' V USMe)}. v(.xt×...×,,xt×.)jliLB3)]AF3- ':' where -5=-iJct6 o-. Then, by (53i and (5s) o-. ' ct2MS ctZMS ''. (56) P9(G);fihl (epE{e),.,-+ (ei-,e),.,,.,±, Fl-Pk(Bo-)]. ' and. hctMt hctM`. S (pl)it,b -I-. (st)i.? +[1-Pk(B,-)]. '. t- ,,', . -. /t (57) ,: ., -, S'''SSIRbk(x.a, ft)-E*pbk(ge, eN)IP*(dxi, ::･,dxh, dfi) ...,･ Gt. '. -<- S ･･･ SSlpgic(xe, 2) - E*e. `ok(6e,., g)lp `*(dx,,..･･･ , dxh, dfi). at-Dbic '' , +S･･･ SSlpSic(xo, x'V)i-- E*phk(6e, 6N)ipk(dx,, ･･･, dx,, dx'"). DSic ' '. ;';g p{･ps(G)+ (i,t, :.;{ Ct(h(Mpi)`,+i) + (hsCl)ij.i -t,-[i-ps(Ba')]p. f. Next, we put ,. FII, ,,,. .i･.,- ..,･,FS=FBt-Gt,.,. ,. ,. - .. ,. '. t a*t((x,,･･･,xh),Al) forL((x,`,･･･,x,t),xN`)EF-,` a9((x,, ･･･,xh), fi) =. .,: . t'Q ,, ..for((xi,･･･,x,t),fit)aii7S. r-t(x. ･･･, x,) = Pt{((g,, ･･･, 6h), gN) E j7i(6i, ･･･, ' 8h) == (xi, ･･･, xh)},. Since FS2FS and FS-FSg- G`, so by (59) we have. 't '/. (58) St･･S(7`(x,,･･･,xh)-rt(xv･･･,xh))P5,.",h)(dxi,･･･,dxh) ''' Xt×･-･×It. '' hci''hct"'`"i'. ;l;llP9(G) ;El (pi)i÷b + <st)ito +[lrPfu(Ba-)]･i:' /;/･t･ ,'-. 't'1' ':.
(18) 18 K. YosHIHARA ' Furthermore,let'. ct={((xS･･･,xS),('oV`); Kl,t ,2,-,ak((xi,''',xh),Ci(('o'")). t/t. , +7t(x,, ･t･,x.) -1 >1(3S}. 'antl. -. . ., ag((x"･･･.･3xh)･4z(a')) '･ ,l,;-i,E-ii",¥.ii'i,lftt)ps. (595' q:･((･Xi,'",Xh),WN)= for((xf,..･,xS),to"'`)GC`. O for((xt,･･･,xh),wN`)ECt.. '. ttt. Then,obviouslywehavethat ' ' O S- qS･((xJ, ･･･, xh), tu'v) $1, A. Kb. (60) O;S,;, qf･((xi, ･･･?xh), di) :;l l-r-"`(xi, ･･･,xh). for all (xS･･･,xZ) and wN` and thati ' (61) 7t(x. ･･･,x,)+ iSa qf･((x. ･･･,x,), wN) '>-- 1-2ps. t=1 for all ((xf, ･･･,xh), di`) ECt.. Now, for any set A` in the h-dimensional Cartesian product of o-algebras SStr × "' ×Sz`. IS' ･･･ S D{ak((xi, ･･･, xh), (;i(('e"))Pk,",h)(dx,, ･･･, dxh). At S S ''･ S PEi,-･,h)(dXi, ･･･, dXh)Sbl,{at.((x. ･･･, x,)･, 4,(w-))}2p'"(d.･-) i. At. I. ' =S･･･S {a9((x,,･･･,x,),S)}2PE,,".,,,xP2kdx,,･･･,dx,,dl). j. L. I [. (AtXi?t)n-R6t , !s 2"t(e)(i'-St)S ･･･ S as,(<x,, ･･･,xh), jg)pEi,".,h) xPgt(d)ci, ･･･,dxh, dfiI). i'. (Atxi?t)n-li'. ' ;$ 2Ht(e)(i+-2a-)S ... S ps(dx,, ... , dx,, dlll) = 2Ut(0)(" g)PE,,..,)(A). Atx .2t. Hence, we have. i. Da"*` ((x,, ･･･ , x,), c,(,t,-)) .,<=, 2Et(e)(i++-,B ). ' foralmostall((xt,･･･,x:).Since･ '･ ･.. l l [ ･1. ,Eak((xi, ･･･ , xh), 4i(di)) = 1-r'6` (xi, ･･･ , xh). for almost all (xf, ･･･,xS), so by TcHEBysHEv's inequality we have. l. i l.
(19) The Shannon Problem for a Unique Transmission Method. 19. p'Vt{ k,.,tL,6, at*((xi, ･･･,xh),cz(di))-(imr-6(xo'",xh)..) > P2} <- K,t .1(ps)2. and consequently ･ .t ' . (62) - '''' ･'p?',,..,,,,xp'vt(c)H<- 2illl`l9.'1iis"si26,7). ,. Thus, if we put . (63) Qt(di)=S･･･S(,Kz..iqt･((x,,･･･,xh),di)+lt(xi,････,xh))p?i,.",h)(dx,,･･･,dxh). .Ntx･..xXt ･. ' -. then, from (58), (61) and (62), we.. have ･ (64) EQt(di)li(i-2ps)[i--2"ii'le,,'((p'Ii)';)]-(p4,)C,t.,･ (sh,f,t.,-[i-pt*(Bs)] .. '. ' ForeachOEeandk(1$k.<.MS).let - '' '. ' a.d ' 'P-O`ic(Xfi'fi)---PSic<Xe'ft)-E*PSk(60,6) ' -N. (6s) v-,(to"")=,IIi3,S･･･Sp-e`k(xe,C'i(w'v))qS･(xi,'",xh,to'")Pfi,･･･,h)(dF,i,'::,dxh). Xtx-･xlt 'l , +S･･･SS p-et,(xfi,x'")pk(dx,,i･･･,dx,,dfi). xtx･･･xxtx.j?t-iiiS Let ptb,(.,,...,.,,...)'='lz?t3,K.ll.ll}3p--h,(.,,e,(.--))ai,t(x,,･･･,xh,ci(to"")). ' '. mhk(xi"･, xh) == E{phetic(xe, C,(to"'))a"*` ((x,, ･･･, xh), C,(di))}. = Sr,abk((Xi, ･'', Xh), fi)p'etic(xo, fi)P3(dN). F. = Sfi(.,;...,.,)a9(<Xi, ''' , Xh), fi)Pe"k(xe, xN)p(dx) .. Since by the definition of F-V. 6`. , for ((xS ･･･,x:), xN`))eFS '. lp-etk(xe, fi)[g P{; k= 1, ･･･,Met; 0 G e, so for almost all (xf, ･･･,xE) G X`× ･･･ × X`, we have P'"`{ l' pteic(xl, '" , xh, di)-meic(x,, ･･･, xh)l> p,} ;:s (P{)2lp2gU/iitll),(,"-ii'). '. (cf. the proof of (62)). Thus we have . P{,,,,,,,,xp'"t{Hb,},:i(g{)2･,73,,`i('lll,(i'g) ...,.....
(20) 20 , ,･ 'K.･YosHIHARA. '. where tis,={(.f,...,xs,.N,):ipts,(x,,...,x,,di)dms,(.,,･･･,x,)il>pg}.'' Next, we remark that S'" S Mhig(Xi, ''' , Xh)PEi, ,h)(dXi, '" , dXh) == S' ''' SS i-oe`ic(xe, Al)Pk(dx, ''･ , dxh, djif). ,ItXny-XXt Pli,6t , -, '. and. ' x")P9(dxi, ･･･,dx,,, dfi) =o. ' 'kC.II;iS..,,P-e`ic(Xe, v. Accordingly, applying the same method used in [3], we have (66) ･/,' Etv-'i(cor)'l:$2,(f?g+pspf+2-(--i(3-f)2HtiOi(i"l)((pl,),+[f[i'/i));,).･ ,･. Finally,weput , ･.･ i. a. st=2Ht(@)'i{,i6-o',･pf=erp,,in(2"t(e)-s2J-,,[1.-pk(B,-)]d+li) '. andfor,.,hbeiiB. iij=2-f(,`(e'ntiOiOll.ps.,=i,'2-"`"e)LroBi,r, ,-. shk(to'") = ,llii S･･･ S phic(xo, c,(di""))qg((x,, ･･･, xh), di)pti, ,h)(dxi, ･･･, dxh). Xtx -･x Xt 1, +i,,.....S,1,,,l9.Ssic.(XO'N)Pk(dXi' 'dxh,dfi)+[1-Q`(to"u)]E.,ps,(6,,g).. Then, we have l. !. -(M-p-it)-t/i'Zt`b- ;-:S'2`H"e'15eb''," r{;g..,x. t;.,a.=i,si,, bElv-et,(tuN)l-s{; 2-"`(e' ioBoo ;. :t ' ....(sM,)t,C.`,;s2-Ht(e)-,a,,--. ･. d. '. Thus,by(64),weobtainthat'forsufncientlylarget ' ･. (67)' ' EQ`(di)l-lll-q$i･'･ ,',･･. where '･･ e･.･ '. ' ' ' '')g$Fi2-Ht(@'io6oo+E17,pg(Bs)]+[1-ps(B,-)]i'2bMtct･,. for which . ,, lim gS･ (vt)a ,. o. " l'. ,t-OO. for any a>O. By (66) and theidefinition of P{ we have. '. j l. r i. I ". I.
(21) The Shannon Problem fot a Unique Transmission Method 21･. (68) maxmaxEISS,(di)-E.pS,(6e,6)l ' '--'' eG.e IS.k-<.Mt o. i:;l {ll.a,X,,l.ll,/LX.,,.El"-'etk(wN')(+ C`(?pM,t)l+lii1) + h(/itt?l,(÷tbC` +P{[1Hpk(BE]]rO,. as t-->oo. Accordingly, we have that by (67). ' p'vt{Q(.--v) lii: i-4g6} ;nv}it .2'.'i.,". ., .4. '. and that by (68) and Lemma 2. P'"t{(se,(to'v),･･･,So.,(di))E[iVh]6 forall0Eiie}lli2. ' exists at least for all sufliciently large t. Thus, we can conclude that' there one tuN ,t G 9t such that. (69) Q`(('v",) lli;il-4{;o3. i. tt. (SS,(wNo),･･･,ShM,(w'V,))Ei[ii7S]6' for'all0Ee.. Now, putting , '･,. (7o) xNt, =cg.(toN,), q:･(x,, ･･･,xh) =qg`(xi,････,xh, dio). wehavethesystemwhiehsatisfiedtheconditiorttf'(i),(ii)and(v). . It remains to show that the conditions (iii) and (iv> are satisfied by the. functionsq2･(x.･-,xh)i=1,･･･,KS, We remark that by the definitions of G`, FS and qg･(xi, ･･･,xh, di). qS･(xi,･･･,xh,di)=O for((xl,･･･,x'.t),4t･(tu")5'GB3 holds. Heuce by Lemma 2 and (70), if qS,(xi, ･･･,xh) >O, then. , lpkic(xe･ fit)-E*e,Ste(6fi, g"')l;Sf3 . .. ., and all 0E e, le == 1, ･･･,MS and i=1, ･･･,KS. ThUs, we have (50). Moreover, '. withthearralogousmethodwecanshowthat ,-･- '' ,,･･'. ･ q!･(x.･･･,xh,to"",)=O for((x:.･･･,xS),C2･(w'"e))EUSk'. '. which implies qS･(x.･･･,xh)==O.-So, if q"-,>, O, then .we have (51). .Thus, we. havethetheorem. , ' ..16. The proof of Theorem 1.. '. ' '/ ''t' At first, let E>O be arbitrarY fixed'ahd choQse,.a,>O 'so small that SEd{g-g and. ; tt ,,,,. '.?Lm.{Iti(ie-,'?-<.Iis6.,,,...,,.,,. ,(7i) ,... ,.,. t.. 1 b'y ･the definiir, holds. ((71) is possible by Assumption II of Theorem 1)...Then,.
(22) 22 K. YosHmARA. '. '. tions of Kat and L3 and ' (71), we have -. 1,. tt. '. ,･ , , }glg {lft -=o..,. Now, we shall use the system obtained in Theorem 3. Let. '. `. t tttt/ttt.tt /. :. q-ot = S ･･･ S r3(xi, ･･･, xh)Pti,...,h)(dx,, ･-･, dxh). '. ' Xtx･-xXt- '`･ ･ q".tt==1-Qt.. Then, by (46) and (47),'`. ". we have '''. ' , K3 t/ (72) OS. q-,C S.1 (i=-1, O, 1, ･･･,KS); Zq-S=:1. tt. pt. i=-1 ･. Here,weput ' " Rs...[ ijzC2LS` ]+1 ,, .(i=:T,,1,O,1,･･･,K$.). (73). K3. ,Re,==L3r,.￡.-,Rg･.- ,,･. '' ' 1 '''' and ,,''/4ti''' "･.'. tt. '. sS-,･(x,, ･･･,xh) = Rs. 48(xi, ･･･txh)(z=i,･･,.I.ir.. S;7=i, ', (i,iQ.. (74) S8j(Xi,''',Xh)=2isr3.(x,,･･･,xh)(i'=i,･･･,Rs). e. seij(xi, ･･･,xh)'= Rlbe, (1'-' Qt(xv t･･,xh))'(7'=1, ･･･,Rk). 11. ' , S!2j(X,,･･･,xh)=O(7'=1,･･･,Re,) . fph6 new 'iunctions sg,･(x,, ･l･,xh) are sk.× ･･･ ×sk-fueasurable. since by. finitions ･. l. I. F l. der. [. ". l. ' ''' ,/e,.if,iill's.,sg･j(le'i',･･･,1,)'pt,,...,h,(dx,,'･･･,d.,)=I/T`iff,oi::-:l:i'･･{}i.s. soifweput '. '. '. l. s. '. 1･. ttt /tt ' ' ･ i,. ,..PS'j=.S.,;l.ftr,sz`j(xi,･･･,xh)pa･,,,.gd¥.･･`,d.,) '. '. l. , , (i.=-2,･･･,KS,y'=1,･･･,RS･). " then we can easily show that the P:･ti satisfy the condition (71) in Theorem We･iremark here that, b-y (50),'(72) and (74)', t:he relations , ･ , , , ... j. i 2.. l l i 1. t :. e. i I.
(23) The Shannon Problem for a Unique Transmission Method. '. '23. '. '. .,=illig, tpt.=,.S,;1.Ilef, hk(xe, x'"i) - E*psk(6e･.eN)iss･j(x,, ･･･, x,)pa, .,,, (dx,, ･･･, dx,)'. '. '. ' xNe) - E*psic(6e, ･eN)Iqg(xi, ･･･', xh)p{i,.1,h) (dxi, ･･i, dxhi ==, t"-iSt･-･ S [psh(xe,. Atx--xlt ,. ;Sl .f''' S IpSk(xe, fit)-E*toSk(6e, 6N)lpE,,...,,,(dx,, ･･･ , dx,) -<. 2'ii,T"`(e). xt×･..×xt. ' ' holds for each l=1, ･･･,KS and for suthciently large t. So, if we put '. ' : . ･.S',';i.I`Obic(Xe,ftii)'--E*iosic(6e,e'")1 ktog･,･,t.= XSf･j(Xi,･･･,xh)PEi,.",h)(dx,,･･･,dxh). '. for l= 1, ･･･,KS. O for l=-2,-1,O. '. then the kph,e. satisfy the condition (18). Accordingly,by Theorem 2, for. all. sufficiently large t, we can find a system {(yS, AS･j),i =-2, ･･･, KS, 7' = 1, ･･･,. RS･}. which is independent of rE I" and has the following properties; (i) for each iand 1', ybe Yt and A;･,ESS, and the Ah are disjoint the same t,. for.. (ii) foreachrEI-'. "' ･' K3R;. (75)･･･' ]E)ZPg･Re(y,,,A,j)llll-6, i== -2 J'=1. (iii) for each rEr K3 R!,. (,II.lll-, Ii.l]=,Pg･jS,,zii(yib y"') Qi(yib dy), ･･･ ,. (76) .. "' ,,.K,2i, tpt.=,P:v'S,,z;ArT(yid,'.:"yV) (?S( yiJ', d.:Ny)) E [ i7I]6. F. '. '. ,, (iv) foreachrer , (77) ' Q"(yt,･,Aij);-}irl---tZP'g. where ･ ･. ZU3=,1.;.PSyr( ;E,rpi ,2r,] -i > g)+2'--s{l7, ct(r). ' '. and. '. (v) foreachrEr <7s)'' ' ,.tlS,,R=l' ,..K2i t,ill.i,kps･,･,,.(?e(y,,l; A,.) -<.(''g' .i. ," (i,,1') JE (t,m) ･. ".
(24) 24 ,K.,YosHmARA . Secondly, for each 0Ee and rE I', we must construct random variables eh,, rp8, ij;, and eNto which take on their values in the measurable sphees (X`, S'. k),. (yt,st.),(9`,s`.-) and(8t,S`.-), respectively. We assume that theSe' random variables are defined on the probability space (9`, E8`, Sl3`), Let 4k,= (6{o,,･･･, 6ito)･. As the initial probability distribution, we use P{i,...,h)(.), so that"fo'r any set. A in the h-dimensional Cartesian product of a-algebras Sk × ･･･ × Sbe. j'. tv ･･' ,' Pt{ek,EiA},.=7P[,l,..,,,(A). ･･ .･ '･''･･,L･:'･･. tt. As the encoding function PS(.,.), we define '' 'PS((Xt','':'XL)''ii)il=,g.,.,ilS.'l'j(.f,i:;",'.Xh)(B`(liSY)'. L. ". The measurability of this function in (xf, ･･･,xk) follows from the measurabi.lity of the functiens sS･,･(x,., :,･･., xh)." For fixed (xg, ･･･,x･%)',PS(.,.) is a prob-. ". ability measu ,re. For e,a,.ch point yg, and each set A`EiSk× ･･･ ×Sk, we have p'Vt{6*, e A, rpo･･= yi,･} = S ･.`i Ssh(xi, ･･･, xh)Pk,･･･ii )(dxi, ''', dxh). /t /t. ' Peg(.:r), for y`G Yt and BN.` E'SS' ' function As th,,g transiSion we Put ,i'. ･/-' ,Pifi(y,B'V,:r),..Qs(y,BN)., ･,.･-,･t,.･.t,,.g･ ' Bytheabo"Ve'definition･･ '.' ･ ,','･ '''' "./. t tt tlt tV tV tV E'". P`{rp-'ro({iiiBlrpo}=Q`r(rpo,B) , .,'.,,, ;,,. holds with probability one. Finally, we shall define the decoding function Pk(Y, A) for yNtG Y` and At,GS`x-. Let X#,be the one obtained in Lemma, 3,. tv tv "v. and put Nt=fi`-i= fit- == fit+. We define. .,,,,,,=,?Ig,rS-1.E.att y,fEAg･j(jl:-/3,,lr.;k',,ig;,,･5,,,g) ' .. Ofor suEA` '. ,,,.. ,1,,,f,.orxNt..g,2ittLte1lj･;I,,.Ag'j,', ...,,.. ". Since the A:･j are disjoint, PE(.,.) is well defined. Frppa the measurability of AS･, we see that PE(.,.) is Measurable as a function of yN`. On the'other.. hand, it is obvious that PE(yN, .) is a probabitl,ity m. easure for each fixed yN` Ei Y`. By. the definition, we have ,, tV -J Pt{6o==NilvNo}=1,....ijof4,G' . :t,'. i. p'Vt{6N,=xN.Iij,}==1, ij6EEOAh' ' ' '. i"'' '. Finally, to prove the theorem we must check that th'e･c6nditioris o･f De-. :. rfinition 6 are satisfied. By construction, it i$ obyious that, for any 0 Ei e and rEI", eeo, rpo, rpNro, 6ny ,.cgnstitgte.ab,. MarkQv chain qnd so the condition (i),is satisfied. By Theorem 2, we obtain that for elach' r G 1, (rp8, ij`To) is connected. I.
(25) The Shannon Problem for a Uni'que Transmission Method 25 by (Q;, VS) and so ･t.he condition (ii) is satisfied..To show that the c'ondition i//ri)la=rel, P.9ti'SKfi,e, d, We reMark ,that if (rp8, rp""rto) is connected by (Qe,. ve,), then. tt. /lt ' /. tVN ,. pt{e,=fi,le., =(x,, ･･･,x,)} ' ''K3･Rg.'･RS'･･ - -'. == Z iZ ]2]sg･j(xi,･･･,xh)Q)i(yib Aim) ' i= -t jL-1 m=1. and for l= -2, -1 and O tv ."fid e*, = (xi, ･･･,xh)} P`{6o=. NN ..=P`{8,=xN+l<l'*o=(Xi,''',Xh)} -t. ' ,.i=ii,KZ.iq.j2R=Sl,,(t.i2tlil-l.iSb9i''"'Xh)`[?i(Y"'A`M), . +s:･d(xi,''',xh)Qi(Yzbli>m,1)i./Aim)) ''. t. .t ttt. hoidforalmostall' (xS･･･,xh). Nowweshallevaluate ,' ''. '. E{phk(6ao, eo) - E*pSic(ee, 6)} -,:.,.li-ilS,i.R.ZI{tK--i.E.IIiQ)i(yikAi-)} ,,,-.. .S,lli. .S.p-, Sic(xo, xNi)sg･j(xi, ･･･ , xh)pc,,..,,,(dx,, ･･･ , d.,). o Re. +[,=¥, .]Et )=,(?e(Yzj, Aim) + (?; O'ij, ]i)`'"-- ,Y. Ai.)]. Xg.11 .S.p-, bic(xe, fa+)s3J(xi, ･ ･, xh)pE,, ,,,(dx,, ･ ･, dx,). K3 Rg.. '. lt .. ==i-¥, ,Z=., {.S,lli.f.p-iic(xe, fiz)sgv(xi, ･･･, xh)pa, ,,, (dx,, ;･., d.,) '. T [1-Qi(yij', Aij)]S'" Sp-teic(xe, fiz)sS･j'(xi, ''' , xh)PCi,･･･,h)(dxi, '" , dxh). xtx･･･×.?rt. '. K3 Rf. + ,;., .￡=,(?i(-Yib A`m)S'''S Pe`ic(xe, xNL)sg･,･(x,, ･･･ , x,). rt.m')fri..f) xtx...x xt ' 'tL. .." tt .t. '. × PC,,...,,,(dx,, ,:･･ .,, dx,). ' +[,l.III],inlil.i,Q'4(yw･Ai-)::t-QS(yw･']i)'m,iy.Ai-)] . ' ''×.S,Jri.S.P-,b'icZXe,xNc)sEj(x2････,xh)pCi,･..,,,(dx,,･l･,dx,).' ', ,.''I'..
(26) 26 ' . 'K.YosHmARA ' '. ' Using (45), (50), (51), (74), (77) and (78), we have the following inequalities:. K3Re･' '. '. ,¥, ,;, [1-Q;(yi,, A,,)] ×.f.'.1;.S.P;bic(Xe, Nt)sS,(xi, ''', xh)Pk,･ ,h)(dxi, ･;.･ , dxh) ;sl`,,6,, ･ IU'S. K3 R:. KS R2 2 2 2 ]E) ([?i(yij, Aim) i= -2 J'=1t=1 m=1 (i,j') iE (t,m). o. '. ×S･･･Slp'o`ic(xo, fi+)1s:･,･(xi, ･･･, xh)p?i,,..,h)(dx,, ･･･, dxh) :g -63-. xt×･,･×.yt. O RC ,.Z,., ,S,). '. si. Q;(yw･, Aij)S ･･･ S lp-e`k(xo, fi+)ls2･,･(x,, ･･･ , x,)pf,,..,,,(dx,, ･･･ , dx,). xtx -･x xt r-f{g S.,'.' i[,- (?`(Xi, '" , xh) + rS (xi, ･･･ , x,)]lp-et,(xe, fi.)I. '. tll.lili. × PE,,...,,,(dx,, ･･･, dx,). ' ,Si, [,=20.,. R.]Sz=, Q;(y,j, Ai.) + (?s(yij, S)'- ,ty. Ai.)] '. × S''' S p"-e` ic(xe, xN+)sS(xi, '" , xh)P(i,...,h)(dxl, ･･･ , dxh). ..tlt× -･× xt. ;;S gbSS･･･Slp-e` ic(xe, N÷)lp h,..,,) (dx,, ･･･,dxh) .-< sb3･(vt) i+ab , JX't× "'× Xt ,.2,Om, t{. Il=,[,t9, .R21,(?;(yij, Atm) + Q)i(yiX ]i)'- ,v,.At.)]. ". ×.i.'.::.S.,l P-e`ic(Xe, i+)IsSj(xi, ･･･, xh)p(i, ,h)(dx,, ･･･, dx,). o L. ;-{S･･･S[1-Q`(xi,･･･,xh)+rS(x.･･･,xh)] .rt×･･-×xt × 1pS,(xo, fi.)lp(,,",,,(dx,, ･･･ , dx,). llsS･･･S[1-Qt(x.･･･,x,)+rs(x.･･･,x,)]･ xtx･･-×xt. , IpSk(xe, N+)IPEi,･･･,i,)(dxts ･･･,dxh), By Lemma 4, ￠S･ Lb6-,t--O as t--÷oo andby Lemmg,,5, ￠3･(vt)-iiF, ->O as t--.c>o.. Furthermore, by Lemma l, z. I.
(27) The Shannon Problem for a Unique Transmission Method. '. '. S ･･･ S [1=Q`(x,, ･･･, xh) +rS(xi, t･･,' xh)]PEk".,h)(dx,, ･･･ , dxh). Itx".xxt ' -. ' t.. =1-pS(F6)+o((vt)-a) ･. iSl,.Z.,aePe`eg( Zi(i60,',i)) -1 >--2--)+2'e"`(0)+o((.t)-a). '. Thus,byLemma5,wehave- ････'-. '. '. '. S･･･ S [1-Qt(x. ･･･, x,)+rS(x,, ･･･, x,)] Xt×･･･xXt. ×lp-e`ic(xe,N+)lPEi,･･･,h)(dXi,'",dXh) ;;l [S ･･･ S [1 - Q`(xi, ･･･ , xh) +r3 (x,, ･･･ , xh)] It × ･･･ × .}r't. XPk...,h)(dx. ･･･, dx,)]-rr+bb- (vt) itb -o. (as t-oo). Combining above results, fgr all suthciently large t, we have. K3. `(79) IEpnye`k(6eo,6No)-,Z.,S'"Sp-etle(Xe,XNi)q:'(Xi,''',Xh). .ir'tx･･･xlt '' '. ･･ -.･ -. XP2,,".,,,(dx,,･･･,dx.)Is6.. On the other hand, by Lemma 1 and Assumption VII S''･SS pbo`ic(xe,fi)p`*(dx.･･･,dx,,dfi). .rtx･･･xtx2t-FS･ '. bl ' ' ･･ ;:;l(1-P`.(F6))T+b'･(ct)-i7+-b--O. '. Since SSic can be rewritten... as. , KS. '. Shic == ,;.,S'" S p` Sic(xe, fii)4E･(x. ･･･ , xh)pE,,,",,)(dx. ･･･ , dx,) v. .irtx m × It. +S･''SS p-otk(Xe,xN)Pt*(dxi,''･,dxh,dN)+E*pSk(6e,8N), .irtx･--×.rtxlt-pS. so for sufficiently large t, the vector (,i.lii 'S'" S pSi(xe, fii)qS･(xi, ･･･, xh)p ti,･･･,h)(dx. ･･･, dxh) + E*ps,(go, gN), ･･･,. (80). .It)<･･･><It .. ' ,il.lli S･･･S p-etMo(xe, xNi)q3(x,, ･･･, xh)pCi, ,h)(dx. ･･･, dxh)+E*pbMe(ge, gny)) xt×-･･×x't .. '. 27.
(28) .. . , K.YosHIHARA. 28. Is an element of [VIZS],. Consequently, we conclude that for all suthciently large t' .. (Ephi(6eo, eo), '" , EpSMe(6eo, 8o)) E [ Wb]e･. J. Thus, we have the theorem.. Acknowledgement The author is indebted to Professor M. UDAGAwA of Tokyo University of Education for his kind guidance, enco'uragementC and helpfu" 1 $uggestions.. References. e. 6. 1) BLAcKwELL, D., L. BREiMAN and A.J. THoMAsiAN: The capacity of a class of channels, Ann. Math. Stat., Vol., 30 (1959) 1223-1241.. 2) DoBRusHiN, R.L.: Allgemeine Formulierung des Shannonshen Hanptsatzes der.Informationstheorie, Arbeiten 2ur Informationstheorie IV (Ubersetzung aus dem Russischen) VEB Deutscher Verlag der Vissenschaften, Berlin (1963). 3) DoBRusHiN, R.L.: Mathematical problems in the Shannon theory of optimal coding of information, Fourth Berle. Sym. on Mathematical Statistics and Probability,. Vol.I(1961). .. 4) FEiNsTEiN, A.: Foundations of injormation theory, New 'York,McGraw-Hill (1958). 5) KoLMoGoRov, A.N.: Theorie der Nachrichtenubermittlung, Arbeiten 2ur Injormationstheorie L 2 Awflage (Ubersetzgng aus dem Russischen) (1961) VEB Deutscher Verlag der Wissenschaften, Berlin. 6) MA Xi-wEN: Feinstein's lemma for finitesystem of transmitters, (in Chinese) Acta Mathematica Sinica, Vol. 14 (1964) 291-303.. 7) PiNsKER, M.S.: Injornzation and information stability of random variables qnd ftrg06C4e)S.SeS, (TranSlated froF,i Russia,? by A. Feinstein) Holden-day Inc., san Francis,.co. 8) WoLFowiTz, J.: qn coding theorems for general sirnultaneous channels. Trans. IRE. Prob. GrouP on Circuit Theory, CT-7, 513-516 (1960). 9) WoLFowiTz, J.: Coding theerems ofinjormation theory, 2nd' ed. Springer-Verltz'g,. o. Berlin (1964).. '. o. G.
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