Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON LIMIT BEHAVIOURS FOR FELLER'S UNFAIR-FAIR-GAME AND ITS RELATED MODEL

  • An, Jun (College of Mathematics and Statistics Chongqing Technology and Business University)
  • Received : 2022.02.05
  • Accepted : 2022.09.26
  • Published : 2022.11.01
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Abstract

Feller introduced an unfair-fair-game in his famous book [3]. In this game, at each trial, player will win 2k yuan with probability pk = 1/2kk(k + 1), k ∈ ℕ, and zero yuan with probability p0 = 1 - Σ∞k=1 pk. Because the expected gain is 1, player must pay one yuan as the entrance fee for each trial. Although this game seemed "fair", Feller [2] proved that when the total trial number n is large enough, player will loss n yuan with its probability approximate 1. So it's an "unfair" game. In this paper, we study in depth its convergence in probability, almost sure convergence and convergence in distribution. Furthermore, we try to take 2k = m to reduce the values of random variables and their corresponding probabilities at the same time, thus a new probability model is introduced, which is called as the related model of Feller's unfair-fair-game. We find out that this new model follows a long-tailed distribution. We obtain its weak law of large numbers, strong law of large numbers and central limit theorem. These results show that their probability limit behaviours of these two models are quite different.

Keywords

Acknowledgement

The author would like to thank the anonymous reviewers for their valuable comments on revision of the manuscript, which have greatly improved the quality of this paper.

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