We show that the accumulated gain Sn and the maximal gain Mn in n St. Petersburg games satisfy almost sure limit theorems with nondegenerate limits, even though ordinary asymptotic distributions do not exist for Sn and Mn with any numerical centering and norming sequences.

The discovery of the almost sure central limit theorem (Brosamler, Math. Proc. Cambridge Philos. Soc. 104 (1988) 561-574; Schatte, Math. Nachr. 137 (1988) 249-256) revealed a new phenomenon in classical central limit theory and has led to an extensive literature in the past decade. In particular,...

We characterize, by means of an integral test, the upper and lower classes associated with the law of the iterated logarithm for subsequences. We also prove strong laws for the frequency of indices in connection with the lower class.

We prove a strong invariance principle for the two-parameter empirical process of stationary sequences under a new weak dependence assumption. We give several applications of our results.

Given a triangular array a={an,k,1[less-than-or-equals, slant]k[less-than-or-equals, slant]kn,n[greater-or-equal, slanted]1} of positive reals, we study the complete convergence property of for triangular arrays of independent random variables. In the Gaussian case we obtain a simple...