J. of Math. (PRC) Shannon-McMillan, , McMillan [2] Breiman [3] , Algoet Cover [10] AEP. P (X n m = x n m) = p m,n (x n m) > 0, x i X, 0 m i n. (1.

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1 Vol. 35 ( 205 ) No. 4 J. of Math. (PRC), (, ) : a.s. Marov Borel-Catelli. : Marov ; Borel-Catelli ; ; ; MR(200) : 60F5 : O2.4; O236 : A : (205) Shao-McMilla,. Shao 948 [],, McMilla [2] Breima [3] L a.s.. Chug [4], Moy [5], Perez [6] Keiffer [7] L. Barro [8] Orey [9], Algoet Cover [0] AEP. ( [0] ).,.,,. (), Marov Borel-Catelli. (X ) N X = {, 2, N}, X m = (X m,, X ), x m = (x m,, x ) X m. (X ) N P (X m = x m) = p m, (x m) > 0, x i X, 0 m i. (.) p m, (x x m ) = P (X = x Xm = x m ), 0 m <. (.2) (a ) N, f a,(ω) =, f a,(ω) X a+ a +. l p(xa+ a + ), ω : : (0704) ; (308085QF3; MA04) ; (202089; ). : (989 ),,,, :.

2 970 Vol. 35 a 0,. X a+ a +,. 2 h a,(x a +) = p (x x a +) l p (x x a +); x X H a,(ω) = h a,(x a +). H a, X X a +. 2 (X ) N (.), [f a,(ω) + =a +2 H a,(ω)] = 0 a.s.. t, Λ () a,(t, ω) = EΛ () a,(t, ω) a+ exp{t l p (X X a+ )} =a+2 a+ =a+2 =E{E[Λ () a,(t, ω) X a+ a + } =E{E[Λ () a, (t, ω) E[e E[e t l p (X X a+ ) X t l pa+(xa+ Xa+ e a+ ) t l pa+(xa+ Xa+ a+ a+ ],, ) X a+ a + a+ =E{Λ () a, (t, ω) E[et l p a+(x a+ Xa+ ) X a+ a + ] t l pa+(xa+ Xa+ E[e a+ ) X a+ a + ] } =E[Λ () a, (t, ω)] = = E[Λ () a,(t, ω)] =. Marov, ɛ > 0, P ( P ( = ] Xa+ a + ]} l Λ() a,(t, ω) ɛ) = P (Λ () a,(t, ω) e ɛ ) EΛ() a,(t, ω) = e ɛ, e ɛ l Λ() a,(t, ω) ɛ) e ɛ <, Borel-Catelli, = P ( l Λ() a,(t, ω) ɛ) = 0. ɛ, l Λ() a,(t, ω) 0 a.s.. (.3)

3 No. 4 : 97 l Λ() = + =a +2 a,(t, ω) {t l p (X X a +) l E[e t l p (X X a+ ) X a +]}, (.4) (.3), (.4) + =a +2 t l p (X X a +) + =a +2 l x x (x > 0) 0 e x x x 2 e x, = t 2 + =a +2 + =a +2 + =a +2 + =a +2 + =a +2 l E[e t l p (X X a+ ) X a +] a.s., {t l p (X X a +) te[l p (X X a +) X a +]} {l E[e t l p (X X a+ ) X a +] te[l p (X X a +) X a +]} {E[e t l p (X X a+ ) X a +] te[l p (X X a +) X a +]} {E[(e t l p (X X a+ ) t l p (X X a +)) X a +]} 0 < t <, (.5) t, t E[l 2 p (X X a +)e t l p (X X a+ ) X a +] a.s.. (.5) + =a +2 + =a +2 max{x t+ l 2 x, x > 0} = 4 e 2 ( t) 2, {l p (X X a +) E[l p (X X a +) X a +]} E[l 2 p (X X a +)e t l p (X X a+ ) X a +] a.s.. E[l 2 p (X X a +)e t l p (X X a+ ) X a +] N = p t (x x a +) l 2 p (x x a +)p (x x a +) = x = N x = 4N e 2 ( t), 2 p t+ (x x a +) l 2 p (x x a +)

4 972 Vol. 35 t (.6), t 0, + =a +2 + =a +2 + =a +2 < t < 0, if + =a +2 (.7) (.8), {l p (X X a +) E[l p (X X a +) X a +]} 4N e 2 ( t) 2 = 4Nt a.s., (.6) e 2 ( t) 2 {l p (X X a +) E[l p (X X a +) X a +]} 0 a.s.. (.7) {l p (X X a +) E[l p (X X a +) X a +]} 0 a.s.. (.8) + =a +2 {l p (X X a +) E[l p (X X a +) X a +]} = 0 a.s.,, Ee l p(xa+) = ɛ > 0, (.9), (.0) + =a +2 N x a+= [l p (X X a +) + H a,(ω)] = 0 a.s.. (.9) e l p(xa+) p(x a+) = N, Marov, P [ l p(x a +) ɛ] N = e ɛ <. = l p(x a +) = 0 a.s.. (.0) l p(x a +) + + =a +2 [l p (X X a +) + H a,(ω)] = 0 a.s., [f a,(ω) + =a +2 H a,(ω)] = 0 a.s..

5 No. 4 : 973 [],, [2 3] []., [] ( a = 0 [] ).. 3 t, M m, (t, X m ) =E[e tp m, (X X m,x ) Xm ] = e tp m, (x x m,x ) p m, (x x m ), 0 m <, (.) x X M m, (t, x m ) Xm = x m, p m, (X Xm ). 2 (a ) N, (X ) N (.), b a, = mi{p a,(x x a ), x i X, a i }, = a +, a + 2,, α > 0, a + =a + Λ (2) a,(t, ω) = + =a + e α/b a, = M <, p a, (X X a ) a + =a + = N e tp a, (X X a ) M a,(t, Xa ). a.s.. (.2), a+ [ =a + EΛ (2) a,(t, ω) =E{E[Λ (2) a,(t, ω) X a+ a ]} =E{E[ a + =a + e tp a, (X X a ) M a,(t, X a =E[Λ (2) a, (t, ω)] = =. tp l Λ(2) a,(t, ω) 0 a (X, X ) a + =a + a ]} ) Xa+ a.s.. l M a,(t, Xa )] 0 a.s..

6 974 Vol. 35 l x x (x > 0) 0 e x x x 2 e x, = t 2 + =a + + =a + + =a + + =a + x = t 2 N + [tp a (X, X ) Nt] a [l M a,(t, X a [M a,(t, X a N N =a + x = + ) Nt] ) Nt] p a,(x Xa )[e tp p b a, =a + a a, (x X a, (x X a )e t p a, (x X a ) ) tp a (x, Xa )] e t b a, a.s.. (.3) 0 < λ <, max{xλ x, x > 0} =. 0 < t < α, e l λ tn tn e(α t) + =a + [p a (X, Xa ) N] + b a, =a + + =a + e t b a, + =a + = tn + b a, =a + e α/b a, = tnm 0, (t 0) a.s., e(α t) ( et e α )/b a, e α/b a, [p a (X, Xa ) N] 0 a.s.. (.4) α < t < 0, (.3) if tn if =tn if + =a + [p a (X, Xa ) N] + b a, =a + + b a, =a + tnm 0, (t 0) a.s., e(α + t) e t b a, e (t+α)/b a, e α/b a,

7 No. 4 : 975 (.4), (.5) if + =a + + =a + [p [p a (X, Xa ) N] 0. (.5) a (X, X ) N] = 0 a.s.,.. ( []) (X ) N (.), α > 0, b = mi{p 0, (x x 0 ), x i x, 0 i },. a e α/b = M <, = {p 0, (X X 0,, X ), m} a.s. N, = p 0, (X X 0 ) = N a.s.. 2 a. 2 m, (X ) N m, α > 0, p(x m 0 ) = P (X m 0 = x m 0 ) > 0, x i X, p m+ (x m+ x +m ) = P (X m+ = x m+ X +m = x +m ) > 0, 0. b = mi{p m+ (x +m x +m ) : x i X, i m + }. a + =a + + =a + e α/b = M <, p m+ (X m+ X a ) = N. 3 (X ) N, p(x 0 ) = P (X 0 = x 0 ) > 0, x 0 X, a.s.. (.6) a + =a + = N a.s.. p (X )

8 976 Vol. 35 [] Shao C E. A mathematical theory of commuicatio[j]. Bell Sgst. Tech. J., 948, 27: [2] McMilla B. The basic theorems of iformatio theory[j]. A. Math. Stat., 953, 24: [3] Breima L. The idividual ergodic theorem of iformatio theory[j]. A. Math. Stat., 957, 28: [4] Chug K L. A ote o the ergodic theorem of iformatio theory[j]. A. Math. Stat., 96, 32: [5] Moy S C. Geeralizatio of the Shao-McMilla theorem[j]. Pacific J. Math., 96, [6] Pierze J R. The early days of iformatio theory[j]. IEEE Tras. If. Theory, 973, 6: 3 8. [7] Kieffer J C. A simple proof of the Moy-Perez geeralizatio of the Shao-McMilla theorem[j]. Pacific J. Math., 974, 5: [8] Barro A R. The strog ergodic theorem for desities: geeralized Shao-McMilla-Breima theorem[j]. A. Prob., 985, 3: [9] Orey S. O the Shao-Perez-Moy theorem[j]. Cotemp. Math., 985, 4: [0] Algoet P, Cover T M. A sadwich proof of the Shao-McMilla-Breima theorem[j]. A. Prob., 988, 6(2): []. [J]., 997, 7(4): [2],. [J]., 2008, 3(4): [3] Shi Zhiya, Yag Weiguo. Some it properties of radom trasitio probability for secod-order ohomogeous Marov chais idexed by a tree[j]. J. Ie. Appl., ID , SOME LIMIT THEOREMS FOR DISCRETE INFORMATION SOURCES JIAN Xu, WANG Zhog-zhi (School of Mathematics Physics Sciece ad Egieerig, Ahui Uiversity of Techology, Maasha , Chia) Abstract: I this paper, we study the properties of geeralized etropy ad the coditioal probability of radom harmoic mea of discrete iformatio sources. By usig Marov s iequality, we put forward a ew approach of studyig strog it theorem, Borel-Catelli lemma ad coditioal momet geeratig fuctio. Keywords: Marov s iequality; Borel-Catelli lemma; geeralized harmoic mea; coditioal momet geeratig fuctio; etropy 200 MR Subject Classificatio: 60F5