## Zentralblatt MATH

Publications of (and about) Paul Erdös Zbl.No:  225.60015
Autor:  Erdös, Paul; Rényi, Alfréd
Title:  On a new law of large numbers. (In English)
Source:  J. Anal. Math. 23, 103-111 (1970).
Review:  We shall prove first (in \S2) the new law of large numbers for the simplest special case, that is for independent repetitions of a fair game. For this special case the theorem can be stated as follows: if the game is played N times, the maximal average gain of a player over [C log2 N] consecutive games (C \geq 1,[x] denotes the integral part of x), tends with probability one to the limit \alpha, where \alpha is the only solution in the interval 0 < \alpha \leq 1 of the equation

1/C = 1- ({1+\alpha \over 2} ) log2 ({2 \over 1+\alpha} ) - ({1- \alpha \over 2} ) log2 ({2 \over 1- \alpha} ).

In \S3 we generalize this result to an arbitrary sequence \etan (n = 1,2, ...) of independent, identically distributed random variables with expectation 0, the common distribution of which satisfies the condition, that its moment-generating function \phi (t) = E(e\etant) exists in an open interval around the origin. We prove that for every \alpha in a certain interval 0 < \alpha < \alpha0 one has

P (limN ––> +oo max0 \leq n \leq N-[C log N] {\etan+1+\etan+2+...+\etan+[C log N] \over [C log N]} = \alpha ) = 1,     (*)

where C = C(\alpha) is defined by the equation e-(1/C) = maxt \phi (t)e- \alpha t. In \S4 we discuss the special case of Gaussian random variables, in which case our result is essentially equivalent to a previous result of P. Lévy about the Brownian movement process. In \S5 we give as an application of the result of \S3, a new proof of the theorem of P. Bártfai on the stochastic geyser problem'', using the fact that the functional dependence between C and \alpha in (*) determines the distribution of the variables uniquely (Theorem 3). The result of \S2 can also be applied in probabilistic number theory; as a matter of fact it was such an application which led the first named author to raise the problem which is solved in the present paper.
Classif.:  * 60F15 Strong limit theorems
60J65 Brownian motion

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