Summary
Let X={X k } be an I.I.D. random sequence and Y={Y k } be a symmetric independent random sequence which is also independent of X. Then X and X+Y={X k +Y k } induce probability measures μ X and μ X+Y on the sequence space, respectively. The problem is to characterize the absolute continuity of μ and μ X+Y and give applications to the absolute continuity of stochastic processes; in particular we give a sufficient condition for the absolute continuity of the sum of Brownian motion and an independent process with respect to the Brownian motion.
We assume that the distribution of X 1 is equivalent to the Lebesgue measure and the density function f satisfies
Under this condition we shall give some sufficient conditions and necessary conditions for μ X ∼μ X+Y . The critical condition is \(\sum\limits_k {\mathbb{E}[|Y_k |^2 :|Y_k | \leqq \varepsilon ]^2 < + \infty }\) for some ɛ>0. In particular in the case where X is Gaussian, we shall give finer results. Finally we shall compare the condition (C) with the Shepp's condition:
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Beckenbach, E.S., Bellman, R.: Inequalities. Berlin Heidelberg New York: Springer 1971
Fernique, X.: Ecole d'Ete de Probabilites de Saint-Flour IV. Lect. Notes Math., vol. 480. Berlin Heidelberg New York: Springer 1974
Ihara, S.: Stochastic processes and entropy. Tokyo: Iwanami 1984 (in Japanese)
Kakutani, S.: On equivalence of infinite product measures. Ann. Math. 49, 214–224 (1948)
Rozanov, Yu.A.: On the density of one Gaussian measure with respect to another. Th. Probab. Appl. 7, 82–87 (1962)
Sato, H.: An ergodic measure on a locally convex topological vector space. J. Func. Anal. 43, 149–165 (1981)
Sato, H.: Characteristic functional of a probability measure absolutely continuous with respect to a Gaussian Radon measure. J. Func. Anal. 61, 222–245 (1985)
Sato, H.: On the convergence of the product of independent random variables. J. Math. Kyoto Univ. 27, 381–385 (1987)
Shepp, L.A.: Distinguishing a sequence of random variables from a translate of itself. Ann. Math. Stat. 36, 1107–1112 (1965)
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Kitada, K., Sato, H. On the absolute continuity of infinite product measure and its convolution. Probab. Th. Rel. Fields 81, 609–627 (1989). https://doi.org/10.1007/BF00367307
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DOI: https://doi.org/10.1007/BF00367307