Absolute continuity of a symmetric three-valued random translation of a Gaussian sequence (Q2752180)

Let \(G=(G_n)\) be a sequence of iid standard Gaussian random variables and \(Y=(Y_n)\) another sequence of independent random variables that is independent of \(G\). Let \(\mu_G\) and \(\mu_{G+Y}\) be distributions in \(R^\infty\) of \(G\) and \(G+Y\), respectively. The following problem is investigated: give necessary and sufficient conditions for equivalence of \(\mu_G\) and \(\mu_{G+Y}\) in terms of \(\mu_Y\), only. When \(Y\) is a deterministic sequence the classical Kakutani theorem gives such a condition. When \(Y_n\) have two-valued distribution for each \(n\), this problem was solved by \textit{H. Mizumachi} and \textit{H. Sato} [in: Trends in probability and related analysis, 233-246 (1997)]. In the present paper the authors find such a necessary and sufficient condition when \(Y_n\) have symmetric three-valued distribution for each \(n\), i.e. \(P(Y_n=\pm a_n) =p_n/2\), \(P(Y_n=0)= 1-p_n\), \(0<p_n<1\), \(n\geq 1\).NEWLINENEWLINEFor the entire collection see [Zbl 0964.00044].