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DOI: 10.46698/w0408-5668-5674-e
On the Rate of Convergence of Ergodic Averages for Functions of Gordin Space
Podvigin, I. V.
Vladikavkaz Mathematical Journal 2024. Vol. 26. Issue 2.
Abstract: For an automorphisms with non-zero Kolmogorov-Sinai 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 automorphism-invariant filtration of \(\sigma\)-algebras \(\mathfrak{F}_n\). A function from the Gordin class is an orthogonal projection with respect to the operator \(I-E(\cdot|\mathfrak{F}_n)\) 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 Voln\'{y}. 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.
Keywords: rates of convergence in ergodic theorems, filtration, martingale method
For citation: Podvigin, I. V. On the Rate of Convergence of Ergodic Averages for Functions of Gordin Space, Vladikavkaz Math. J., 2024, vol. 26, no. 2, pp. 95-102.
DOI 10.46698/w0408-5668-5674-e
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