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Kachurovskii, Alexander Grigoryevich

 Statistics Math-Net.Ru Total publications: 17 Scientific articles: 17 Presentations: 5

 Number of views: This page: 3146 Abstract pages: 7235 Full texts: 2624 References: 615
Doctor of physico-mathematical sciences (2000)
Speciality: 01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date: 21.01.1961
E-mail:
Keywords: ergodic theory, rates of convergence in ergodic theorems, Fejer sums, unifications of ergodic theorems and martingale convergence theorems
UDC: 517.987, 519.214, 519.216
MSC: 28D, 37A, 60F, 60G

Subject:

It was proved (1996) that a power rate of convergence in von Neumanns ergodic theorem is equivalent to the power (with the same exponent) singularity at zero point of a spectral measure of averaging function with respect to the dynamical system. I.e. it was shown that the estimates of convergence rates in this ergodic theorem are necessarily the spectral ones. Estimates of the rates of convergence were obtained (1996; since 2010 – with the students): in von Neumanns ergodic theorem – via the singularity at zero point of the spectral measure, and via the speed of decay of correlations (i.e., the Fourier coefficients of this measure); in Birkhoffs ergodic theorem – via the rate of convergence in von Neumanns theorem, and via a speed of decay of probabilities of large deviations. Asymptotically exact estimates of the rates of convergence are obtained in both these ergodic theorems: for certain well-known billiards, and Anosov systems.

It was shown (1998) that both ergodic averages and martingales can be viewed as particular degenerate cases of the one new general class of stochastic processes; convergences of this new general process a.e. (an extra condition of integrability of the supremum of module of the process was omitted by the student I.V. Podvigin in 2010) and in the norm, are proved – and both maximal and dominant estimates take place, too.

It turns out (2018), that the Fejer sums for measures on the circle and the norms of the deviations from the limit in the von Neumann ergodic theorem both are calculating, in fact, with the same formulas (by integrating of the Fejer kernels) --- and so, this ergodic theorem is a statement about the asymptotics of the growth of the Fejer sums at zero for the corresponding spectral measure. As a result, availablle in the literature, numerous estimates for the deviations of Fejer sums at a point allowed to obtain new estimates for the rate of convergence in this ergodic theorem.

Biography

Graduated from Faculty of Mathematics and Mechanics of Novosibirsk State University in 1983 (Department of Mathematical Analysis). Ph.D thesis was defended in 1987 at Sobolev Institute of Mathematics, Novosibirsk. D.Sci thesis was defended in 1999 at Steklov Mathematical Institute at St. Petersburg. Principal place of work since 1983 – Sobolev Institute of Mathematics (with an interruption in 1997–1999 for the doctorate at Steklov Mathematical Institute at St. Petersburg).

Main publications:
1. Kachurovskii A. G., “Rates of convergence in ergodic theorems”, Russian Math. Surveys, 51:4 (1996), 653–703
2. Kachurovskii A. G., “General theories unifying ergodic averages and martingales”, Proc. Steklov Inst. Math., 256 (2007), 160–187
3. Kachurovskii A. G., Sedalishchev V. V., “Constants in estimates for the rates of convergence in von Neumann’s and Birkhoff’s ergodic theorems”, Sb. Math., 202:8 (2011), 1105–1125
4. Kachurovskii A. G., Podvigin I. V., “Large deviations and the rate of convergence in the Birkhoff ergodic theorem”, Math. Notes, 94:4 (2013), 524–531
5. Kachurovskii A. G., Podvigin I. V., “Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems”, Trans. Moscow Math. Soc., 77 (2016), 1–53

http://www.mathnet.ru/eng/person17305
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Publications in Math-Net.Ru
 2019 1. A. G. Kachurovskii, I. V. Podvigin, “Measuring the Rate of Convergence in the Birkhoff Ergodic Theorem”, Mat. Zametki, 106:1 (2019),  40–52    ; Math. Notes, 106:1 (2019), 52–62 2018 2. A. G. Kachurovskii, “The Fejer integrals and the von Neumann ergodic theorem with continuous time”, Zap. Nauchn. Sem. POMI, 474 (2018),  171–182 2017 3. A. G. Kachurovskiĭ, I. V. Podvigin, “Large deviations of the ergodic averages: from Hölder continuity to continuity almost everywhere”, Mat. Tr., 20:1 (2017),  97–120    ; Siberian Adv. Math., 28:1 (2018), 23–38 2016 4. A. G. Kachurovskii, I. V. Podvigin, “Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems”, Tr. Mosk. Mat. Obs., 77:1 (2016),  1–66    ; Trans. Moscow Math. Soc., 77 (2016), 1–53 2013 5. A. G. Kachurovskii, I. V. Podvigin, “Large Deviations and the Rate of Convergence in the Birkhoff Ergodic Theorem”, Mat. Zametki, 94:4 (2013),  569–577        ; Math. Notes, 94:4 (2013), 524–531 2012 6. A. G. Kachurovskii, V. V. Sedalishchev, “On the Constants in the Estimates of the Rate of Convergence in the Birkhoff Ergodic Theorem”, Mat. Zametki, 91:4 (2012),  624–628      ; Math. Notes, 91:4 (2012), 582–587 2011 7. A. G. Kachurovskii, V. V. Sedalishchev, “Constants in estimates for the rates of convergence in von Neumann's and Birkhoff's ergodic theorems”, Mat. Sb., 202:8 (2011),  21–40        ; Sb. Math., 202:8 (2011), 1105–1125 8. N. A. Dzhulaĭ, A. G. Kachurovskiĭ, “Constants in the estimates of the rate of convergence in von Neumann's ergodic theorem with continuous time”, Sibirsk. Mat. Zh., 52:5 (2011),  1039–1052    ; Siberian Math. J., 52:5 (2011), 824–835 2010 9. A. G. Kachurovskii, V. V. Sedalishchev, “On the Constants in the Estimates of the Rate of Convergence in von Neumann's Ergodic Theorem”, Mat. Zametki, 87:5 (2010),  756–763      ; Math. Notes, 87:5 (2010), 720–727 10. A. G. Kachurovskii, A. V. Reshetenko, “On the rate of convergence in von Neumann's ergodic theorem with continuous time”, Mat. Sb., 201:4 (2010),  25–32        ; Sb. Math., 201:4 (2010), 493–500 2007 11. A. G. Kachurovskii, “General Theories Unifying Ergodic Averages and Martingales”, Tr. Mat. Inst. Steklova, 256 (2007),  172–200        ; Proc. Steklov Inst. Math., 256 (2007), 160–187 2006 12. A. G. Kachurovskii, “The entropy brick of an automorphism of a Lebesgue space”, Mat. Zametki, 80:6 (2006),  943–945        ; Math. Notes, 80:6 (2006), 885–887 1999 13. A. G. Kachurovskii, “Convergence of averages in the ergodic theorem for groups \$\mathbb Z^d\$”, Zap. Nauchn. Sem. POMI, 256 (1999),  121–128        ; J. Math. Sci. (New York), 107:5 (2001), 4231–4236 1998 14. A. G. Kachurovskii, “Martingale ergodic theorem”, Mat. Zametki, 64:2 (1998),  311–314      ; Math. Notes, 64:2 (1998), 266–269 1996 15. A. G. Kachurovskii, “The rate of convergence in ergodic theorems”, Uspekhi Mat. Nauk, 51:4(310) (1996),  73–124        ; Russian Math. Surveys, 51:4 (1996), 653–703 1992 16. A. G. Kachurovskii, “Time fluctuations in the statistical ergodic theorem”, Mat. Zametki, 52:1 (1992),  146–148      ; Math. Notes, 52:1 (1992), 744–745 1991 17. A. G. Kachurovskii, “Fluctuation of averages in the strong law of large numbers”, Mat. Zametki, 50:5 (1991),  151–153      ; Math. Notes, 50:5 (1991), 1202–1203

Presentations in Math-Net.Ru
 1 Ñóììû Ôåéåðà è ýðãîäè÷åñêàÿ òåîðåìà ôîí ÍåéìàíàA. G. Kachurovskii Seminar on Probability Theory and Mathematical StatisticsOctober 12, 2018 18:00 2 Ñóììû Ôåéåðà ïåðèîäè÷åñêèõ ìåð è ýðãîäè÷åñêàÿ òåîðåìà ôîí ÍåéìàíàA. G. Kachurovskii Dynamical systems and differential equationsOctober 8, 2018 18:30 3 Rates of convergence in ergodic theorems for Anosov diffeomorphismsAlexander Kachurovskii International Conference “Anosov Systems and Modern Dynamics” dedicated to the 80th anniversary of Dmitry AnosovDecember 23, 2016 15:40 4 Deviations of Fejer sums and rates of convergence in the von Neumann ergodic theoremA. G. Kachurovskii International Conference "Geometric Analysis and Control Theory"December 8, 2016 15:00 5 Îöåíêè ñêîðîñòåé ñõîäèìîñòè â ýðãîäè÷åñêèõ òåîðåìàõ ôîí Íåéìàíà è ÁèðêãîôàA. G. Kachurovskii Seminar on Probability Theory and Mathematical StatisticsOctober 4, 2013 20:00

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