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Limit Theorems for a Generalized ST Petersburg Game

Part of: Stochastic processes Limit theorems Real functions

Published online by Cambridge University Press:  14 July 2016

Allan Gut
Allan Gut*
Affiliation:
Uppsala University
*
∗Postal address: Department of Mathematics, Uppsala University, Box 480, SE-751 06 Uppsala, Sweden. Email address: allan.gut@math.uu.se
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Abstract

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The topic of the present paper is a generalized St Petersburg game in which the distribution of the payoff X is given by P(X = sr(k-1)/α) = pqk-1, k = 1, 2,…, where p + q = 1, s = 1 / p, r = 1 / q, and 0 < α ≤ 1. For the case in which α = 1, we extend Feller's classical weak law and Martin-Löf's theorem on convergence in distribution along the 2n-subsequence. The analog for 0 < α < 1 turns out to converge in distribution to an asymmetric stable law with index α. Finally, some limit theorems for polynomial and geometric size total gains, as well as for extremes, are given.

Keywords

St Petersburg gamesums of i.i.d. random variablesFeller's weak law of large numbersdomains of attractionconvergence along subsequencesextremesstable lawslow variationregular variation

MSC classification

Secondary: 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks 26A12: Rate of growth of functions, orders of infinity, slowly varying functions
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 3 , September 2010 , pp. 752 - 760
Copyright
Copyright © Applied Probability Trust 2010 

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