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**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 log_{2} 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} **)** log_{2} **(**{2 \over 1+\alpha} **)** - **(**{1- \alpha \over 2} **)** log_{2} **(**{2 \over 1- \alpha} **)**. In \S3 we generalize this result to an arbitrary sequence \eta_{n} (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 < \alpha_{0} one has

P **(****lim**_{N ––> +oo} **max**_{0 \leq n \leq N-[C log N]} {\eta_{n+1}+\eta_{n+2}+...+\eta_{n+[C log N]} \over [C log N]} = \alpha **)** = 1, (*) where C = C(\alpha) is defined by the equation e^{-(1/C)} = **max**_{t} \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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