THE LIMITING DISTRIBUTION OF THE ST-PETERSBURG GAME

The St. Petersburg game is a well-known example of a random variable which has infinite expectation. Csorgo 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), then a limiting distribution exists exactly when the fractional parts of 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.