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On the rate of convergence of the St. Petersburg game. (English) Zbl 1240.60011

Let P(X n =2 k )=2 -k with independent X n , X n (c) =min(X n ,c), S n (c) = k=1 n X k (c) . γ n (c) γ means either γ(1/2,1), γ n γ or γ=1 and γ n has no other limit points than 1 and 1/2; {x}, [x] denote the fractional and the integer parts of x. The authors prove the following theorems.

1. For ε>0, P(|S n (n) -E(S n (n) )|(nlog 2 n) -1 >ε)<2n 4-(loglogn)εlog 2 e .

2. n -1 i=1 n (X i -n) + , n=n k , converges in distribution for k to some nondegenerate limit if and only if n k /2 [log 2 n k ] (c) γ, and the limit has the characteristic function exp( 0 (e itx -1-itx(1+x 2 ) -1 )d(-2 {log 2 [γ(x+1)]} (x+1) -1 )).

3. Same for n k -1 S n k (n k ) -log 2 n k with -x -1 2 {log 2 (γx)} for x<1, 0 otherwise, under d.

4. (VarS n c n ) -1/2 (S n c n -E(S n c n )) tends in distribution to the standard normal one if and only if c n /n0.

5. (0·16+o(1))/log 2 nE(log 2 (S n /(nlog 2 n)))-log 2 log 2 n/(log2)(log 2 n)(2·52+o(1))/log 2 n.

6. E((log 2 (S n /(nlog 2 n))) 2 )=O(1/logn).

Theorem 5 appears in a paragraph entitled “Growth rate of sequential St. Petersburg portfolio games” in which some results of L. Györfi and P. Kevei [Algorithmic learning theory. Proceedings. Berlin: Springer. Lecture Notes in Computer Science 5809. Lecture Notes in Artificial Intelligence, 83–96 (2009; Zbl 1262.91047)] are presented. The paper finishes by showing histograms of some log 2 S n , log 2 S n (c) and by proving that log 2 S n is not asymptotically normal.

MSC:
60E05General theory of probability distributions
60F15Strong limit theorems
60G50Sums of independent random variables; random walks
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