Journal of the
Korean Mathematical Society JKMS

Feller introduced an unfair-fair-game in his famous book \cite{Feller-1968}. In this game, at each trial, player will win $2^k$ yuan with probability $p_k=1/2^kk(k+1)$, $k\in \mathbb{N}$, and zero yuan with probability $p_0=1-\sum_{k=1}^{\infty}p_k$. Because the expected gain is 1, player must pay one yuan as the entrance fee for each trial. Although this game seemed ``fair", Feller \cite{Feller-1945} 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 $2^k=m$ 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: Feller's unfair-fair-game, probability limit behaviour, long-tailed distribution, central limit theorem

MSC numbers: Primary 60F05, 60G50

Supported by: This work was financially supported by Key Platform Open Projects of Chongqing Technology and Business University (Grant no. KFJJ2018099); Natural Science Foundation of Chongqing, China (Grant no. cstc2020jcyj-msxmX0328); The Science and Technology Research Program of Chongqing Municipal Education Commission (Grant no. KJQN202000838).