The almost sure limit theorem for sums of random vectors (Q1603208)

The paper investigates conditions for pathwise (or almost sure) limit theorems for sums of independent random vectors \((\xi_j)_{j\geq 1}\) with values in a separable normed space. Let \(Z_k:=\frac{1}{B_k}\sum^k_{j=1}\xi_j-A_k\) denote the normalized partial sums. The author gives sufficient conditions under which weak convergence of \(Z_k\) implies convergence of the empirical measures \(Q_n:=\frac{1}{\gamma_n}\sum^n_{k=2} b_k\delta_{Z_k}\), with \(b_k\) positive constants and \(\gamma_n=\sum^n_{k=1}b_k\). This extends earlier work by e.g. \textit{I. Berkes} and the reviewer [Ann. Probab. 21, No.~3, 1640-1670 (1993; Zbl 0785.60014)]. By means of a counterexample, the author shows that a moment condition in that paper is almost optimal.