On almost sure limit theorems

By a sequence of random vectors $\{\zeta_k\}$, we can construct empirical distributions of the type $ Q_n = (\log n)^{-1} \sum_{k=1}^n \delta_{\zeta_k}/k$. Statements on the convergence of these or similar distributions with probability 1 to a limit distribution are called almost sure theorems. We propose several methods which permit us to easily deduce the almost sure limit theorems from the classical limit theorems, prove the invariance principle of the type “almost sure” and investigate the convergence of generalized moments. Unlike the majority of the preceding papers, where only convergence to the normal law is considered, our results may be applied in the case of limit distributions of the general type.