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On a new law of large numbers. (English) Zbl 0225.60015
We shall prove first (in §2) 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 \[ {1 \over C}=1- \left({1+ \alpha \over 2} \right) \log_2 \left({2 \over 1+ \alpha} \right) - \left({1- \alpha \over 2} \right) \log_2 \left({2 \over 1- \alpha} \right). \] In §3 we generalize this result to an arbitrary sequence \(\eta_n\) \((n=1,2, \ldots)\) of independent, identically distributed random variables with expectation \(0\), the common distribution of which satisfies the condition, that its moment-generating function \(\varphi (t)=E(e^{\eta_nt})\) exists in an open interval around the origin. We prove that for every \(\alpha\) in a certain interval \(0< \alpha < \alpha_0\) one has \[ P \left(\lim_{N \to + \infty} \max_{0 \leq n \leq N-[C \log N]} {\eta_{n+1}+ \eta_{n+2}+ \ldots + \eta_{n+[C \log N]} \over [C \log N]}= \alpha \right) =1, \tag{*} \] where \(C=C(\alpha)\) is defined by the equation \(e^{-(1/C)}= \min_t \varphi (t)e^{- \alpha t}\). In §4 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 §5 we give as an application of the result of §3, 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 §2 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.

60F15 Strong limit theorems
60J65 Brownian motion
Full Text: DOI
[1] H. Cramér,Sur un nouveau théorème-limite de la théorie des probabilités. Actualités Scientifiques et Industrielles, No 736, Hermann et Cie, Paris, 1938.
[2] R. R. Bahadur and R. Ranga Rao, On deviations of the sample mean.Annals of Mathematical Statistics,31 (1960), 1015–1027. · Zbl 0101.12603
[3] P. Lévy,Théorie de l’addition des variables aléatoires indépendantes, Paris, Gauthier-Villars.
[4] P. Bártfai, Die Bestimmung der zu einem wiederkehrenden Prozess gehörenden Verteilungsfunktion aus den mit Fehlern behafteten Daten einer einzigen Realisation.Studia Scientiarum Mathematicarum Hungarica,1 (1966), 161–168. · Zbl 0156.39102
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