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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Random ergodic sequences on LCA groups

Author: Jakob I. Reich
Journal: Trans. Amer. Math. Soc. 265 (1981), 59-68
MSC: Primary 60B15
MathSciNet review: 607107
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Abstract: Let $ {\{ X(t,\omega )\} _{t \in {{\mathbf{R}}^ + }}}$ be a stochastic process on a locally compact abelian group $ G$, which has independent stationary increments. We show that under mild restrictions on $ G$ and $ \{ X(t,\omega )\} $ the random families of probability measures

$\displaystyle {\mu _T}( \cdot ,\omega ) = B_T^{ - 1}\int\limits_0^T {f(t){x_{( \cdot )}}} (X(t,\omega ))dt\quad {\text{for}}\;T > 0{\text{,}}$

where $ f(t)$ is a function from $ {{\mathbf{R}}^ + }$ to $ {{\mathbf{R}}^ + }$ of polynomial growth and $ {B_T} = \int_0^T {f(t)} \;dt$, converge weakly to Haar measure of the Bohr compactification of $ G$. As a consequence we obtain mean and individual ergodic theorems and asymptotic occupancy times for these processes.

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Keywords: Random family of probability measures, stationary stochastic process with independent increments
Article copyright: © Copyright 1981 American Mathematical Society