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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 

References

Adler, A. (1990). Generalized one-sided laws of the iterated logarithm for random variables barely with or without finite mean. J. Theoret. Prob. 3, 587–597.Google Scholar
Breiman, L. (1992). Probability. SIAM, Philadelphia, PA.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Cambridge University Press.Google Scholar
Durrett, R. (2010). Probability: Theory and Examples, 4th edn. Cambridge University Press.Google Scholar
Feller, W. (1957). An Introduction to Probability Theory and Its Applications, Vol. I, 2nd edn. John Wiley, New York.Google Scholar
Grimmett, G. and Stirzaker, D. (2001). Probability and Random Processes, 3rd edn. Oxford University Press, New York.Google Scholar
Gut, A. (2004). An extension of the Kolmogorov–Feller weak law of large numbers with an application to the St. Petersburg game. J. Theoret. Prob. 17, 769–779.Google Scholar
Gut, A. (2005). Probability: A Graduate Course. Springer, New York.Google Scholar
Gut, A. (2010). Limit theorems for a generalized St. Petersburg game. J. Appl. Prob. 47, 752–760.CrossRefGoogle Scholar
Hu, Y. and Nyrhinen, H. (2004). Large deviations view points for heavy-tailed random walks. J. Theoret. Prob. 17, 761–768.CrossRefGoogle Scholar
Maller, R. A. (1978). Relative stability and the strong law of large numbers. Z. Wahrscheinlichkeitsth. 43, 141–148.Google Scholar
Martin-Löf, A. (1985). A limit theorem which clarifies the ‘Petersburg paradox’. J. Appl. Prob. 22, 634–643.Google Scholar
Nolan, J. (2012). Stable Distributions: Models for Heavy Tailed Data. Birkhäuser, Boston, MA.Google Scholar
Norris, J. R. (1997). Markov Chains. Cambridge University Press.Google Scholar
Rényi, A. (2007). Foundations of Probability. Dover Publications.Google Scholar
Stoica, G. (2008). Large gains in the St. Petersburg game. C. R. Math. Acad. Sci. Paris 346, 563–566.CrossRefGoogle Scholar
Zolotarev, V. M. (1986). One-Dimensional Stable Distributions. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar