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A limit theorem which clarifies the ”Petersburg Paradox”. (English) Zbl 0574.60032
Remember the famous ”Petersburg Paradox”. Let \(S_ N\) be the total gain in N games. In his book ”An introduction to probability theory and its applications”, Vol. 1 (1968; Zbl 0155.231), W. Feller proves a modified law of large numbers for it: \(S_ N/N\) Log \(N\to 1\) in probability (Log N denotes the binary logarithm of N). The present author obtains a weak convergence theorem for \(S_ N\) when \(N=2^ n\). His result is as follows: as \(n\to \infty\) \((N=2^ n)\) \((S_ N-NLog N)/N\) has an infinitely divisible limit distribution G(x) with the characteristic function \[ g(u)=\sum^{0}_{-\infty}(\exp (iu2^ k)-1- iu2^ k)2^{-k}+\sum^{\infty}_{1}(\exp (iu2^ k)-1)2^{-k}. \] The tail of G(x) has an asymptotic behaviour: \(1-G(2^ m)\simeq (1.8)2^{-m}\) as \(m\to \infty\). From this the reader can deduce some interesting facts about the ”Petersburg Paradox”.
Reviewer: H.Takahata

60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
60E07 Infinitely divisible distributions; stable distributions
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