|
Zapiski Nauchnykh Seminarov POMI, 1997, Volume 244, Pages 61–72
(Mi znsl511)
|
|
|
|
This article is cited in 2 scientific papers (total in 2 papers)
Double extensions of dynamical systems and a construction of mixing filtrations
M. I. Gordin St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
Abstract:
Let $T$ be an automorphism of a probability space $(X,\mathscr F,P)$ and let $A_s$ and $A_u$ be generators of symmetric Markov transition semigroups on $X$. $A_s$ and $A_u$ are supposed also to be “eigenvectors” for $T$ with eigenvalues $\theta^{-1}$ and $\theta$ for some $\theta>1$. We give a probabilistic construction (based on $A_s$ and $A_u$) of an extension of the quadruple $(X,\mathscr F,P,T)$. This extension $(X',\mathscr F',P,T')$ is naturally supplied with decreasing and increasing filtrations.
Under the assumptions that $A_s$ and $A_u$ commute and that their sum $A_s+A_u$ is bounded below apart from zero we establish very strong decay to zero of the maximal correlation coefficient between the
$\sigma$-fields of these filtrations.
As an application we prove the following assertion under the above conjectures. Let $f\in L_2$ has integral 0 with respect to $P$ and be such that
$$
\sum_{k\ge 0}\bigl((|f|^2_2-|\mathbf P_s(\theta^{-k})f|^2_2)^{1/2}+(|f|^2_2-|\mathbf P_u(\theta^{-k})f|^2_2)^{1/2}\bigr)<\infty.
$$
Then the sequence $\{f\circ T^k, k\in\mathbb Z\}$ satisfies the Functional Central Limit Theorem.
As an example we consider hyperbolic toral automorphisms.
Received: 10.12.1997
Citation:
M. I. Gordin, “Double extensions of dynamical systems and a construction of mixing filtrations”, Probability and statistics. Part 2, Zap. Nauchn. Sem. POMI, 244, POMI, St. Petersburg, 1997, 61–72; J. Math. Sci. (New York), 99:2 (2000), 1053–1060
Linking options:
https://www.mathnet.ru/eng/znsl511 https://www.mathnet.ru/eng/znsl/v244/p61
|
Statistics & downloads: |
Abstract page: | 142 | Full-text PDF : | 53 |
|