Some Integral Inequalities and Absolute Continuity of a Symmetrical Random Translation

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= {Xk} with the distribution ƒ(x) dx and Y = {Yk} is an independent and symmetric random sequence also independent of X, such that Y ∈ l4, a.s.. then X and X + Y = { Xk + Tk } 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 Fields81 (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].