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Limit Theorems for a Generalized Feller Game

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Keisuke Matsumoto  and
Toshio Nakata
Keisuke Matsumoto*
Affiliation:
Fukuoka University of Education
Toshio Nakata*
Affiliation:
Fukuoka University of Education
*
∗ Postal address: Department of Mathematics, Fukuoka University of Education, Akama-Bunkyomachi, Munakata, Fukuoka, 811-4192, Japan.
∗ Postal address: Department of Mathematics, Fukuoka University of Education, Akama-Bunkyomachi, Munakata, Fukuoka, 811-4192, Japan.
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Abstract

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In this paper we study limit theorems for the Feller game which is constructed from one-dimensional simple symmetric random walks, and corresponds to the St. Petersburg game. Motivated by a generalization of the St. Petersburg game which was investigated by Gut (2010), we generalize the Feller game by introducing the parameter α. We investigate limit distributions of the generalized Feller game corresponding to the results of Gut. Firstly, we give the weak law of large numbers for α=1. Moreover, for 0<α≤1, we have convergence in distribution to a stable law with index α. Finally, some limit theorems for a polynomial size and a geometric size deviation are given.

Keywords

Feller's gamesimple symmetric random walklaw of large numbersstable lawextremes

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 54 - 63
Copyright
Copyright © Applied Probability Trust 2013 

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