
On the rate of convergence of ergodic averages for functions of Gordin space
I. V. Podvigin^{} ^{} Sobolev Institute of Mathematics of the Siberian Branch of the RAS, 4 Ac. Koptug Ave., Novosibirsk 630090, Russia
Abstract:
For an automorphisms with nonzero KolmogorovSinai entropy, a new class of $L_2$functions called the Gordin space is considered. This space is the linear span of Gordin classes constructed by some automorphisminvariant filtration of $\sigma$algebras $\mathfrak{F}_n$. A function from the Gordin class is an orthogonal projection with respect to the operator $IE(\cdot\mathfrak{F}_n)$ of some $\mathfrak{F}_m$measurable function. After Gordin's work on the use of the martingale method to prove the central limit theorem, this construction was developed in the works of Volný. In this review article we consider this construction in ergodic theory. It is shown that the rate of convergence of ergodic averages in the $L_2$ norm for functions from the Gordin space is simply calculated and is $\mathcal{O}(\frac{1}{\sqrt{n}}).$ It is also shown that the Gordin space is a dense set of the first Baire category in ${L_2(\Omega,\mathfrak{F},\mu)\ominus L_2(\Omega,\Pi(T,\mathfrak{F}),\mu)},$ where $\Pi(T,\mathfrak{F})$ is the Pinsker $\sigma$algebra.
Key words:
rates of convergence in ergodic theorems, filtration, martingale method.
Received: 21.12.2023
Citation:
I. V. Podvigin, “On the rate of convergence of ergodic averages for functions of Gordin space”, Vladikavkaz. Mat. Zh., 26:2 (2024), 95–102
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https://www.mathnet.ru/eng/vmj912 https://www.mathnet.ru/eng/vmj/v26/i2/p95

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