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By B. Gyires

ISBN-10: 0444853332

ISBN-13: 9780444853332

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Additional resources for Analytic Function Methods in Probability Theory: Colloquium Proceedings (Colloquia mathematica Societatis Janos Bolyai ; 21)

Example text

Let be an information function. 4) =\ If then in [O,IJ 0 if xE(O,I) if x=O or x=1 is the Shannon information function. J. ACZEL and Z. 84): S Is the only nonnegative information function? Since the information function is the measure of information determined by the probability distribution I-x} {x, (xE[ 0, I]) it is natural to suppose the norlnegativity of this measure. , The result of Z. DAR6CZY and I. KATAI has been generalized by G. DIDERRICH [5J in another direction: bounded in [0, I] If the information function then f (x) =S (x) for f is x E [ 0, I] • In this paper we prove two essential results which solve the problem of J.

J. :J. the identity zeroelement. :J. t] / ' j (dy). r n (x-y), in the case when x By our assumption the sequence converges weakly. rxll n ..... + oo r > O. :J. xII r /'j({x:lI[\ xII n r=r(£) clearly r=r(£)). 2 we obtain the following estimations. :J. xII > c) < £ for all n. 1 we find that (2. 10) > £) is any probability measure. :J. :J. :J. yll >£ n r ~ ". 0 if E is sufficiently small large). 7) 2 k + - ,',k )]ll r n n - as r - +00, +00 II n{x:U~ xU ~ E} then tends n r E > O. Thus we find that the conditions for any (iv) 0 ,"k (repeated limit), since (ii) and n(dt)~ n 2 The left hand side tends to 0 correspondingly E.

5) we get c 1 (n). 3. {A Let } +00 (r= 1 ,2, •.. ) n,r n=1 of probability measures on i 2 tending to positive integers lim sup k lim (i) r--+ oo n"'+ oo lim sup k lim (ii) r"'+ oo n"'+ oo lim sup k lim (iii) r"'+ oo for n"'+ oo a n - +00. as +00 J A J t. n lit. yl>e: n,r r n lit. yiISe: r r YA sequence of If 0, (dy) (dy)O 0, n,r U,\ yl12A J n lit. ylSe: r (dy) r n,r n(dy) ° ° A lit. yll>n n,r n ..... + oo r"'+ oo "'k J lim sup lim r n > 0. any PROOF. Let put {k} n e: > 0, then any (i v) for and be sequences *0 A n,r e.

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Analytic Function Methods in Probability Theory: Colloquium Proceedings (Colloquia mathematica Societatis Janos Bolyai ; 21) by B. Gyires

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