Distinguishing a Sequence of Random Variables from a Random Translate of Itself

Abstract

Let $\mathbf{X} = \{X_k\}$ be an i.i.d. real random sequence, let $\epsilon = \{\epsilon_k\}$ be a Rademacher sequence independent of $\mathbf{X}$ and let $\mathbf{a} = \{a_k\}$ be a deterministic real sequence. The aim of this paper is to prove that the mutual absolute continuity of probability measures induced by $\{X_k\}$ and $\{X_k + a_k\epsilon_k\}$ implies $\mathbf{a} \in \ell_4$. This is a generalization of a result of Shepp.