The limiting distribution of the St. Petersburg game

Abstract

The St. Petersburg game is a well-known example of a random variable which has infinite expectation. Csörgő and Dodunekova have recently shown that the accumulated winnings do not have a limiting distribution, but that if measurements are taken at a subsequence b n {b_n} , then a limiting distribution exists exactly when the fractional parts of log 2 b n {\log _2}{b_n} approach a limit. In this paper the characteristic functions of these distributions are computed explicitly and found to be continuous, self-similar, nowhere differentiable functions.