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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

##### MSC:
 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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