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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.

References [Enhancements On Off] (What's this?)

  • [1] J. R. Blum and R. Cogburn, On ergodic sequences of measures, Proc. Amer. Math. Soc. 51 (1975), 359-365. MR 0372529 (51:8736)
  • [2] J. R. Blum and B. Eisenberg, Generalized summing sequences and the mean ergodic theorem, Proc. Amer. Math. Soc. 42 (1974), 423-429. MR 0330412 (48:8749)
  • [3] J. I. Reich, Ph. D. Thesis, Univ. of New Mexico, 1976.
  • [4] W. Rudin, Fourier analysis on groups, Interscience, New York, 1962. MR 0152834 (27:2808)

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

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