Limit Theorems for a Generalized Feller Game

Abstract

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.

Limit Theorems for Super-heavy Tailed Random Variables with Truncation: Application to the Super-petersburg Game

“…(ii) Conditions of Eqs. (18) and (22) for O-super-heavy tailed distributions correspond to [24,Eqs. (13) and 23] for power laws with index α, respectively.…”

“…There are many results of limit theorems for iid random variables of this type distribution, in particular it is interesting when α = 1 (see [1], [2], [3], [22], [25], [26] and references therein).…”

105.48 Approximations for the Feller games

“…This game is written in Feller's textbook (see [2, Section X.1, p. 246]). Hence Matsumoto and Nakata (see [3]) called it the Feller game. The distribution of is X…”

“…Hence it is not so easy to investigate the limit distribution of compared to . Details are discussed in [3].…”

Some limit theorems for weighted negative quadrant dependent random variables with infinite mean

“…The theorem states the condition of the mean’s existence is not necessary, and St. Petersburg game (see [2]) and Feller game (see [3]), which are well known as the typical examples, are formulated by a nonnegative random variable X with the tail probability for each fixed , where denotes Nakata [4] considered truncated random variables and studied strong laws of large numbers and central limit theorems in this situation. In [5], Nakata studied the weak laws of large numbers for weighted independent random variables with the tail probability (1.2) and explored the case that the decay order of the tail probability is −1.…”