SLLN for weighted independent identically distributed random variables
A sequence of complex numbers \({\{a_n}\}\) is called \(q\)-Besicovitch (\(1\leq q<{\infty}\)) if it is in the closure of the set of trigonometric polynomials, in the semi-norm \({\|{\{b_k\}}\|}_{W_q}=[\limsup {1\over n}\sum_{k=1}^{n}\| b_k| ^q]^{1/q}\). A 1-Besicovitch sequence is simply called Besicovitch, and every \(q\)-Besicovitch sequence is Besicovitch. The authors show that if \({\|{{a_k}}\|}_{W_q}<{\infty}\) for some \(q>1\), then \({1\over n}\sum_{k=1}^{n}{a_k}{X_k}\) converges to 0 a.s. for every \({\{X_n}\}\) i.i.d. with \({E(| X_{1}| )<\infty}\) and \(E(X_{1})=0\), and that the result fails for \(q=1\). Moreover, for every i.i.d. sequence \({\{X_n}\}\), with \({X_1}\) nonnegative and not essentially bounded, there is a 1-Besicovitch sequence \({\{a_n}\}\) such that \({1\over n}\sum_{k=1}^{n}{a_k}{X_k}\) diverges almost surely.
scientific article; zbMATH DE number 2083465