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On the absolute continuity of infinite product measure and its convolution
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  • Published: May 1989

On the absolute continuity of infinite product measure and its convolution

  • Kazuhiro Kitada1 &
  • Hiroshi Sato1 

Probability Theory and Related Fields volume 81, pages 609–627 (1989)Cite this article

  • 130 Accesses

  • 14 Citations

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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

$$(C){\text{ }}\int\limits_{ - \infty }^{ + \infty } {\frac{{f''(x)^2 }}{{f(x)}}} dx < + \infty .$$

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:

$${\text{ }}(A){\text{ }}\int\limits_{{\text{ - }}\infty }^{{\text{ + }}\infty } {\frac{{f'(x)^2 }}{{f(x)}}dx < + \infty .}$$

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Authors and Affiliations

  1. Department of Mathematics, Kyushu University, 33 Hakozaki, 812, Fukuoka, Japan

    Kazuhiro Kitada & Hiroshi Sato

Authors
  1. Kazuhiro Kitada
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  2. Hiroshi Sato
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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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  • Received: 02 June 1987

  • Revised: 29 August 1988

  • Issue Date: May 1989

  • DOI: https://doi.org/10.1007/BF00367307

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Keywords

  • Density Function
  • Stochastic Process
  • Convolution
  • Brownian Motion
  • Probability Theory
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