Strong convergence of weighted sums of random elements through the equivalence of sequences of distributions
The authors provide a proof of the SLLN in separable Banach spaces in the general framework of pairwise independent random elements and consider almost sure convergence to 0, as \(n\to \infty\), of weighted sums, \(\sum_{k\geq 1}a_{nk}X_ k\), of random elements in a separable Banach space, where \(\{a_{nk}\); n,k\(\geq 1\}\) is a Toeplitz sequence of real numbers. The basic results in the paper are proved by using an extension due to Blackwell and Dubins of the Skorokhod representation theorem applied to weakly equivalent sequences of probability measures.
weakly equivalent sequences of probability measures
scientific article; zbMATH DE number 4054678
scientific article