Limit behaviors for a β-mixing sequence in the St. Petersburg game

We consider a sequence of non-negative β-mixing random variables {X, Xn : n ≥ 1} from the classical St. Petersburg game.The accumulated gains Sn = X 1 + X 2 + • • • + Xn in the St. Petersburg game are studied, and the large deviations and the weak law of large numbers of Sn are obtained.P -→ 1 as n → ∞.Chow and Robbins [3] showed that S n n log 2 n a.s. 1 as n → ∞, and the set of limit points of S n /n log 2 n is the interval [1, ∞).Csörgő and Simons [4] proved that, with probability one, S n is asymptotic to n log 2 n if the m