On the absolute continuity of infinite product measure and its convolution
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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 μX+Y on the sequence space, respectively. The problem is to characterize the absolute continuity of μ and μX+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 X1 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 ∼μX+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:
$${\text{ }}(A){\text{ }}\int\limits_{{\text{ - }}\infty }^{{\text{ + }}\infty } {\frac{{f'(x)^2 }}{{f(x)}}dx < + \infty .}$$
Keywords
Density Function Stochastic Process Convolution Brownian Motion Probability Theory
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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© Springer-Verlag 1989