Geodesic
Measuring the Rate of Convergence in the Birkhoff Ergodic Theorem
Matematičeskie zametki, Tome 106 (2019) no. 1, pp. 40-52.

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Estimates of the rate of convergence in the Birkhoff ergodic theorem which hold almost everywhere are considered. For the action of an ergodic automorphism, the existence of such estimates is proved, their structure is studied, and unimprovability questions are considered.
Keywords: individual ergodic theorem, rate of convergence in ergodic theorems, unimprovability of estimates, return time, lattice. 
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A. G. Kachurovskii; I. V. Podvigin. Measuring the Rate of Convergence in the Birkhoff Ergodic Theorem. Matematičeskie zametki, Tome 106 (2019) no. 1, pp. 40-52. http://geodesic.mathdoc.fr/item/MZM_2019_106_1_a3/

[1] A. G. Kachurovskii, “Skorosti skhodimosti v ergodicheskikh teoremakh”, UMN, 51:4 (310) (1996), 73–124 | DOI | MR | Zbl

[2] V. F. Gaposhkin, “O skorosti ubyvaniya veroyatnostei $\varepsilon$-uklonenii srednikh statsionarnykh protsessov”, Matem. zametki, 64:3 (1998), 366–372 | DOI | MR | Zbl

[3] V. V. Sedalischev, “Svyaz skorostei skhodimosti v ergodicheskikh teoremakh fon Neimana i Birkgofa v $L_p$”, Sib. matem. zhurn., 55:2 (2014), 412–426 | MR

[4] A. G. Kachurovskii, I. V. Podvigin, “Bolshie ukloneniya i skorosti skhodimosti v ergodicheskoi teoreme Birkgofa”, Matem. zametki, 94:4 (2013), 569–577 | DOI | MR | Zbl

[5] I. V. Podvigin, “O skorosti skhodimosti v individualnoi ergodicheskoi teoreme dlya deistvii polugrupp”, Matem. tr., 18:2 (2015), 93–111 | DOI | MR | Zbl

[6] A. G. Kachurovskii, I. V. Podvigin, “Otsenki skorostei skhodimosti v ergodicheskikh teoremakh fon Neimana i Birkgofa”, Tr. MMO, 77, no. 1, MTsNMO, M., 2016, 1–66

[7] N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publ., Amsterdam, 1958 | MR | Zbl

[8] V. F. Gaposhkin, “O zavisimosti skorosti skhodimosti v usilennom zakone bolshikh chisel dlya statsionarnykh protsessov ot skorosti ubyvaniya korrelyatsionnoi funktsii”, Teoriya veroyatn. i ee primen., 26:4 (1981), 720–733 | MR | Zbl

[9] C. Cuny, “Pointwise ergodic theorems with rate with applications to limit theorems for stationary processes”, Stoch. Dyn., 11:1 (2011), 135–155 | DOI | MR | Zbl

[10] C. Cuny, “Some optimal pointwise ergodic theorems with rate”, C. R. Math. Acad. Sci. Paris, 347:15-16 (2009), 953–958 | DOI | MR | Zbl

[11] L. Keipers, G. Niderreiter, Ravnomernoe raspredelenie posledovatelnostei, Mir, M., 1985 | MR | Zbl

[12] P. Liardet, D. Volný, “Sums of continuous and differentiable functions in dynamical systems”, Israel J. Math., 98 (1997), 29–60 | DOI | MR | Zbl

[13] G. Halász, “Remarks on the remainder in Birkhoff's ergodic theorem”, Acta Math. Acad. Sci. Hungar., 28:3-4 (1976), 389–395 | DOI | MR | Zbl

[14] G. Atkinson, “Recurrence of co-cycles and random walks”, J. London Math. Soc. (2), 13:3 (1976), 486–488 | DOI | MR | Zbl

[15] K. Schmidt, “On reccurence”, Z. Wahrscheinlichkeitstheorie verw Gebiete, 68:1 (1984), 75–95 | DOI | Zbl

[16] I. P. Kornfeld, Ya. G. Sinai, S. V. Fomin, Ergodicheskaya teoriya, Nauka, M., 1980 | MR

[17] U. Krengel, “On the speed of convergence in the ergodic theorem”, Monatsh. Math., 86 (1978), 3–6 | DOI | MR | Zbl

[18] A. del Junco, J. Rosenblatt, “Counterexamples in ergodic theory and number theory”, Math. Ann., 245:3 (1979), 185–197 | DOI | MR | Zbl

[19] J.-C. Yoccoz, Petits diviseurs en dimension 1, Astérisque, 231, Soc. Math. France, Paris, 1995 | MR | Zbl

[20] U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985 | MR | Zbl