On weak convergence in dynamical systems to self-similar processes with spectral representation
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- by Michael T. Lacey
- Trans. Amer. Math. Soc. 328 (1991), 767-778
- DOI: https://doi.org/10.1090/S0002-9947-1991-1066446-5
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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 \[ A_m^{ - 1}{S_{[mt]}}f \stackrel {d}{\Rightarrow } Y(t),\] the convergence being understood in the sense of weak convergence of all finite dimensional distributions in $t$.References
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Bibliographic Information
- © Copyright 1991 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 328 (1991), 767-778
- MSC: Primary 60F17; Secondary 28D05, 60F05
- DOI: https://doi.org/10.1090/S0002-9947-1991-1066446-5
- MathSciNet review: 1066446