Abstract
The aim of this paper is to prove the inequality [formula] for a probability density function ƒ( x), to prove some equalities and inequalities among the related integrals, and then, as an application, to prove that, if ∫ +∞ −∞(ƒ″( x) 2/ƒ( x)) dx < ∞, an IID sequence X= { X k } with the distribution ƒ( x) dx and Y = { Y k } is an independent and symmetric random sequence also independent of X, such that Y ∈ l 4, a.s.. then X and X + Y = { X k + T k } induce mutually absolutely continuous probability measures on the sequence space. These results improve those in K. Kitada and H. Sato [On the absolute continuity of infinite product measure and its convolution, Probab. Theory Related Fields 81 (1989), 609-627] and generalize the problem of L. A. Shepp [Distinguishing a sequence of random variables from a translate of itself, Ann. Math. Statist. 36 (1965), 1107-1112].
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