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Transactions of the American Mathematical Society

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



On weak convergence in dynamical systems to self-similar processes with spectral representation

Author: Michael T. Lacey
Journal: Trans. Amer. Math. Soc. 328 (1991), 767-778
MSC: Primary 60F17; Secondary 28D05, 60F05
MathSciNet review: 1066446
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Abstract: Let $ (X,\mu,T)$ be an aperiodic dynamical system. Set $ {S_m}f = f + \cdots + f \circ {T^{m - 1}}$, where $ f$ is a measurable function on $ X$. Let $ Y(t)$ be one of a class of self-similar process with a "nice" spectral representation, for instance, either a fractional Brownian motion, a Hermite process, or a harmonizable fractional stable motion. We show that there is an $ f$ on $ X$, and constants $ {A_m} \to + \infty $ so that

$\displaystyle A_m^{ - 1}{S_{[mt]}}f\mathop \Rightarrow \limits^d Y(t),$

the convergence being understood in the sense of weak convergence of all finite dimensional distributions in $ t$.

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Keywords: Aperiodic dynamical system, weak convergence, almost sure invariance principle, spectral representation, self-similar, fractional Brownian motion, Hermite processes, harmonizable fractional stable motion
Article copyright: © Copyright 1991 American Mathematical Society