On the universal a.s. central limit theorem (Q950290)

Let \(X_1 ,X_2 ,\dots\) be independent random variables such that for some measurable functions \(g_l \) the weak limit theorem \(g_l (X_1 ,\dots,X_l ) \Rightarrow G\) holds with some distribution function \(G\). The paper gives conditions for the validity of relation \(D_N^{ - 1} \sum\limits_{k = 1}^N {d_k f(g_k (X_1 ,\dots,X_k ))} =\int_{ - \infty }^\infty {f(x)\,dG(x)} \) a.s. The second result covers the case when \(g_k \) depend not on an independent sequence \((X_k )\), but on a more general stochastic process with independent increments.