Almost sure limit theorems for random sums of multiindex random variables
Denote by \(\mathop \to \limits^d \) the convergence in distribution, by \(\mathop \to \limits^p \) the convergence in probability, by \(\mathop \to \limits^w \) the weak convergence of measures, and by \({\mu _\varsigma }\) the distribution of the random variable \(\varsigma \). Let \({\varsigma _n}\) be a sequence of random variables defined on the probability space \((\Omega ,A,P)\). Consider the measures \({Q_n}(\omega ) = {Q_n}(({\varsigma _n}))(\omega ) = \frac{1}{{\log n}}\sum_{k = 1}^n {\frac{1}{k}{\delta _{{\varsigma _k}(\omega )}}} \), where \(\omega \in \Omega \), and \({\delta _x}\) is the point mass at \(x\). Classical limit theorems deal with the convergence \({\varsigma _n}\mathop \to \limits^d \varsigma \) as \(n \to \infty \). In many cases, this convergence implies the convergence of measures \({Q_n}(\omega )\mathop \to \limits^w {\mu _\varsigma }\) as \(n \to \infty \) for almost all \(\omega \in \Omega \). Such limit theorems are called almost sure versions of ordinary limit theorems. In this paper, the author obtains a functional limit theorem for multiindex random sums and its almost sure version.
scientific article; zbMATH DE number 5626717