On the equivalence of certain product measures (Q1326288)

Let \(G=\{g_ k, k \in\mathbb{N}\}\) be a sequence of independent Gaussian centred reduced random variables; let moreover \(Y=\{y_ k, k \in \mathbb{N}\}\) be a sequence independent of \(G\) of independent random variables. For obtaining conditions characterizing the equivalence of the distributions of \(G\) and \(G+Y\), we use the zero-one laws verified by \(G\), first for the convergence of the series \(\sum \sigma_ k g_ k\) or \(\sum \sigma_ k (g^ 2_ k-a_ k)\), secondly for the asymptotic behavior of the sequence \(\{g_ k, k \in \mathbb{N}\}\) and we analyze their maximal effects on the distribution of \(Y\).