On almost sure limit theorems (Q2711119)

Let \( \xi_1,\dots ,\xi_n \) be random vectors and assume that their properly normalized sum \( \zeta_n \) converges weakly. Let \( G \) be the limit distribution. Define the empirical distributions \( Q_n \) by NEWLINE\[NEWLINE Q_n(A)= \frac{1}{\gamma(n)} \sum^n_{k=1}\frac{1}{k}I_A(\zeta_k) NEWLINE\]NEWLINE where \( \gamma(n)= \sum^n_{k=1}\frac{1}{k} \). Under ``almost sure'' type limit theorem for \( \zeta_n \) one understands here that NEWLINE\[NEWLINE P(Q_n \Longrightarrow G) = 1. \tag \(*\) NEWLINE\]NEWLINE Denote by \( \Delta_n(t) \) the difference between the characteristic functions of the distributions \( G \) and \( Q_n \). Already in the introduction the authors prove Theorem 1.1 which states that the assumption \( E|\Delta_n(t) |\rightarrow 0 \) is necessary and the assumption \( \sup_{|t |\leq r}\sum_n \frac{E|\Delta_n(t) |^2}{n\log n} < \infty \) (\( r \) arbitrary) is sufficient for the validity of \((*)\). On the base of Theorem 1.1 almost sure type limit theorems are obtained in the cases when: \( \xi_n \) are independent or weakly dependent vectors (Section 2); \(\zeta_n \) are random elements in a linear metric space (Section 3). Finally, a.s. convergence of generalized functions and moments is investigated in the last Section 4.