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Volumn 51, Issue 4, 1996, Pages 653-703

The rate of convergence in ergodic theorems

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EID: 0030353041     PISSN: 00360279     EISSN: 14684829     Source Type: Journal    
DOI: 10.1070/RM1996v051n04ABEH002964     Document Type: Article
Times cited : (110)

References (63)
  • 2
    • 85164656876 scopus 로고    scopus 로고
    • Pitman, Boston, MA-London 1981
    • N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, Vol. 1, 3rd ed., Vishcha Shkola, Kharkov 1977; English transl., Pitman, Boston, MA-London 1981.
  • 3
    • 0004189013 scopus 로고
    • Fizmatgiz, Moscow
    • N. K. Bari [Bary], Trigonometric series, Fizmatgiz, Moscow 1961; English transl., A treatise on trigonometric series, Pergamon Press, Macmillan Co., New York 1964.
    • (1961) Trigonometric Series
    • Bari, N.K.1
  • 4
    • 0004210126 scopus 로고
    • Pergamon Press, Macmillan Co., New York
    • N. K. Bari [Bary], Trigonometric series, Fizmatgiz, Moscow 1961; English transl., A treatise on trigonometric series, Pergamon Press, Macmillan Co., New York 1964.
    • (1964) A Treatise on Trigonometric Series
  • 5
    • 85002068159 scopus 로고
    • An example of a process with mixing
    • Yu. K. Belyaev, "An example of a process with mixing", Teor. Veroyatnost. i Primenen. 6 (1961), 101-103; English transl. in Theory Probab. Appl. 6 (1964).
    • (1961) Teor. Veroyatnost. i Primenen. , vol.6 , pp. 101-103
    • Belyaev, Yu.K.1
  • 6
    • 85164640967 scopus 로고
    • Yu. K. Belyaev, "An example of a process with mixing", Teor. Veroyatnost. i Primenen. 6 (1961), 101-103; English transl. in Theory Probab. Appl. 6 (1964).
    • (1964) Theory Probab. Appl. , vol.6
  • 7
    • 0038926356 scopus 로고
    • On the dependence of the rate of convergence in the strong law of large numbers for stationary processes on the rate of decay of the correlation function
    • V. F. Gaposhkin, "On the dependence of the rate of convergence in the strong law of large numbers for stationary processes on the rate of decay of the correlation function", Teor. Veroyatnost. i Primenen. 26 (1981), 720-733; English transl. in Theory Probab. Appl. 26 (1981).
    • (1981) Teor. Veroyatnost. i Primenen , vol.26 , pp. 720-733
    • Gaposhkin, V.F.1
  • 8
    • 85164636934 scopus 로고
    • V. F. Gaposhkin, "On the dependence of the rate of convergence in the strong law of large numbers for stationary processes on the rate of decay of the correlation function", Teor. Veroyatnost. i Primenen. 26 (1981), 720-733; English transl. in Theory Probab. Appl. 26 (1981).
    • (1981) Theory Probab. Appl. , vol.26
  • 11
    • 0003520148 scopus 로고
    • Wolters-NoordhofF Publishing, Groningen
    • I. A. Ibragimov and Yu. V. Linnik, Independent and stationarily connected variables, Nauka, Moscow 1965; English transl., Independent and stationary dependent sequences of random variables, Wolters-NoordhofF Publishing, Groningen 1971.
    • (1971) Independent and Stationary Dependent Sequences of Random Variables
  • 12
    • 0040704196 scopus 로고    scopus 로고
    • Oscillations of averages in the ergodic theorem
    • V. V. Ivanov, "Oscillations of averages in the ergodic theorem", Dokl. Akad. Nauk 347 (1996), 736-738; English transl. in Russian Acad. Sci. Dokl. Math. 53 (1996).
    • (1996) Dokl. Akad. Nauk , vol.347 , pp. 736-738
    • Ivanov, V.V.1
  • 13
    • 85164648673 scopus 로고    scopus 로고
    • V. V. Ivanov, "Oscillations of averages in the ergodic theorem", Dokl. Akad. Nauk 347 (1996), 736-738; English transl. in Russian Acad. Sci. Dokl. Math. 53 (1996).
    • (1996) In Russian Acad. Sci. Dokl. Math. , vol.53
  • 14
    • 0039518925 scopus 로고
    • Boundedness of fluctuation of means in the Birkhoff-Khintchine ergodic theorem
    • A. G. Kachurovskii, "Boundedness of fluctuation of means in the Birkhoff-Khintchine ergodic theorem", Dokl. Akad. Nauk SSSR 315 (1990), 530-532; English transl. in Soviet Math. Dokl. 42 (1991).
    • (1990) Dokl. Akad. Nauk SSSR , vol.315 , pp. 530-532
    • Kachurovskii, A.G.1
  • 15
    • 85164638544 scopus 로고
    • A. G. Kachurovskii, "Boundedness of fluctuation of means in the Birkhoff-Khintchine ergodic theorem", Dokl. Akad. Nauk SSSR 315 (1990), 530-532; English transl. in Soviet Math. Dokl. 42 (1991).
    • (1991) Soviet Math. Dokl. , vol.42
  • 16
    • 0038926421 scopus 로고
    • A fluctuation ergodic theorem
    • A. G. Kachurovskii, "A fluctuation ergodic theorem", Dokl. Akad. Nauk SSSR 317 (1991), 823-826; English transl. in Soviet Math. Dokl. 43 (1991).
    • (1991) Dokl. Akad. Nauk SSSR , vol.317 , pp. 823-826
    • Kachurovskii, A.G.1
  • 17
    • 85164643225 scopus 로고
    • A. G. Kachurovskii, "A fluctuation ergodic theorem", Dokl. Akad. Nauk SSSR 317 (1991), 823-826; English transl. in Soviet Math. Dokl. 43 (1991).
    • (1991) Soviet Math. Dokl. , vol.43
  • 18
    • 34249836316 scopus 로고
    • Fluctuations of averages in the strong law of large numbers
    • English transl. in Math. Notes 50 (1991)
    • A. G. Kachurovskii, "Fluctuations of averages in the strong law of large numbers", Mat. Zametki 50:5 (1991), 151-153; English transl. in Math. Notes 50 (1991).
    • (1991) Mat. Zametki , vol.50 , Issue.5 , pp. 151-153
    • Kachurovskii, A.G.1
  • 19
    • 85164646647 scopus 로고
    • Fluctuation of means in the Birkhoff-Khintchine ergodic theorem
    • A. G. Kachurovskii, "Fluctuation of means in the Birkhoff-Khintchine ergodic theorem", Trudy Inst. Mat. (Novosibirsk) 21 (1992), 52-86; English transl. in Siberian Adv. Math. 3:2 (1993).
    • (1992) Trudy Inst. Mat. (Novosibirsk) , vol.21 , pp. 52-86
    • Kachurovskii, A.G.1
  • 20
    • 85164632673 scopus 로고
    • A. G. Kachurovskii, "Fluctuation of means in the Birkhoff-Khintchine ergodic theorem", Trudy Inst. Mat. (Novosibirsk) 21 (1992), 52-86; English transl. in Siberian Adv. Math. 3:2 (1993).
    • (1993) Siberian Adv. Math. , vol.3 , Issue.2
  • 21
    • 85164651114 scopus 로고
    • A. G. Kachurovskii, "Fluctuation of means in the Birkhoff-Khintchine ergodic theorem", Trudy Inst. Mat. (Novosibirsk) 21 (1992), 52-86; English transl. in Siberian Adv. Math. 3:2 (1993).
    • (1993) Siberian Adv. Math. , vol.3 , Issue.2
  • 23
    • 0001945762 scopus 로고
    • On the dispersion of the time averages of a stationary stochastic process
    • V. P. Leonov, "On the dispersion of the time averages of a stationary stochastic process", Teor. Veroyatnost. i Primenen. 6 (1961), 93-101; English transl. in Theory Probab. Appl. 6 (1961).
    • (1961) Teor. Veroyatnost. i Primenen. , vol.6 , pp. 93-101
    • Leonov, V.P.1
  • 24
    • 85164641714 scopus 로고
    • V. P. Leonov, "On the dispersion of the time averages of a stationary stochastic process", Teor. Veroyatnost. i Primenen. 6 (1961), 93-101; English transl. in Theory Probab. Appl. 6 (1961).
    • (1961) Theory Probab. Appl. , vol.6
  • 25
    • 85164636857 scopus 로고
    • V. P. Leonov, "On the dispersion of the time averages of a stationary stochastic process", Teor. Veroyatnost. i Primenen. 6 (1961), 93-101; English transl. in Theory Probab. Appl. 6 (1961).
    • (1961) Theory Probab. Appl. , vol.6
  • 27
    • 0040110586 scopus 로고
    • The strong law of large numbers for a stationary sequence
    • V. V. Petrov, "The strong law of large numbers for a stationary sequence", Dokl. Akad. Nauk SSSR 213 (1973), 42-44; English transl. in Soviet Math. Dokl. 213 (1973).
    • (1973) Dokl. Akad. Nauk SSSR , vol.213 , pp. 42-44
    • Petrov, V.V.1
  • 28
    • 85164645723 scopus 로고
    • V. V. Petrov, "The strong law of large numbers for a stationary sequence", Dokl. Akad. Nauk SSSR 213 (1973), 42-44; English transl. in Soviet Math. Dokl. 213 (1973).
    • (1973) Soviet Math. Dokl. , vol.213
  • 29
    • 0008577530 scopus 로고
    • Selected topics from the metric theory of dynamical systems
    • V. A. Rokhlin, "Selected topics from the metric theory of dynamical systems", Uspekhi Mat. Nauk 4:2 (1949), 57-128; English transl. in AMS Transl. (2) 49.
    • (1949) Uspekhi Mat. Nauk , vol.4 , Issue.2 , pp. 57-128
    • Rokhlin, V.A.1
  • 30
    • 85164640039 scopus 로고    scopus 로고
    • V. A. Rokhlin, "Selected topics from the metric theory of dynamical systems", Uspekhi Mat. Nauk 4:2 (1949), 57-128; English transl. in AMS Transl. (2) 49.
    • AMS Transl. , vol.49 , Issue.2
  • 33
    • 0003266562 scopus 로고
    • Fluctuations in martingales
    • A. A. Chashka, "Fluctuations in martingales", Uspekhi Mat. Nauk 49:2 (1994), 179-180; English transl. in Russian Math. Surveys 49:2 (1994).
    • (1994) Uspekhi Mat. Nauk , vol.49 , Issue.2 , pp. 179-180
    • Chashka, A.A.1
  • 34
    • 85164639568 scopus 로고
    • A. A. Chashka, "Fluctuations in martingales", Uspekhi Mat. Nauk 49:2 (1994), 179-180; English transl. in Russian Math. Surveys 49:2 (1994).
    • (1994) Russian Math. Surveys , vol.49 , Issue.2
  • 35
    • 85164652765 scopus 로고
    • A. A. Chashka, "Fluctuations in martingales", Uspekhi Mat. Nauk 49:2 (1994), 179-180; English transl. in Russian Math. Surveys 49:2 (1994).
    • (1994) Russian Math. Surveys , vol.49 , Issue.2
  • 38
    • 0010940115 scopus 로고
    • Exponential convergence rates for the law of large numbers
    • L. E. Baum, M. Katz, and R. R. Read, "Exponential convergence rates for the law of large numbers", Trans. Amer. Math. Soc. 102 (1962), 187-199.
    • (1962) Trans. Amer. Math. Soc. , vol.102 , pp. 187-199
    • Baum, L.E.1    Katz, M.2    Read, R.R.3
  • 39
    • 84968512234 scopus 로고
    • Convergence rates in the law of large numbers
    • L. E. Baum and M. Katz, "Convergence rates in the law of large numbers", Trans. Amer. Math. Soc. 120 (1965), 108-123.
    • (1965) Trans. Amer. Math. Soc. , vol.120 , pp. 108-123
    • Baum, L.E.1    Katz, M.2
  • 40
    • 0000808359 scopus 로고
    • Variants of the law of the iterated logarithm
    • N. H. Bingham, "Variants of the law of the iterated logarithm", Bull. London Math. Soc. 18 (1986), 433-467.
    • (1986) Bull. London Math. Soc. , vol.18 , pp. 433-467
    • Bingham, N.H.1
  • 41
    • 0039518926 scopus 로고
    • An upcrossing inequality with applications
    • E. Bishop, "An upcrossing inequality with applications", Michigan Math. J. 13 (1966), 1-13.
    • (1966) Michigan Math. J. , vol.13 , pp. 1-13
    • Bishop, E.1
  • 42
    • 0038926352 scopus 로고
    • A constructive ergodic theorem
    • E. Bishop, "A constructive ergodic theorem", J. Math. Mech. 17 (1968), 631-639.
    • (1968) J. Math. Mech. , vol.17 , pp. 631-639
    • Bishop, E.1
  • 44
    • 27144512485 scopus 로고
    • Remarks on the remainder in Birkhoff's ergodic theorem
    • G. Halasz, "Remarks on the remainder in Birkhoff's ergodic theorem", Acta Math. Hungar. 28 (1976), 389-395.
    • (1976) Acta Math. Hungar. , vol.28 , pp. 389-395
    • Halasz, G.1
  • 45
    • 0040704200 scopus 로고
    • Martingale formulation of ergodic theorems
    • M. Jerison, "Martingale formulation of ergodic theorems", Proc. Amer. Math. Soc. 10 (1959), 531-539.
    • (1959) Proc. Amer. Math. Soc. , vol.10 , pp. 531-539
    • Jerison, M.1
  • 46
    • 0039640851 scopus 로고
    • Inequalities for the ergodic maximal function
    • R. L. Jones, "Inequalities for the ergodic maximal function", Studia Math. 60:2 (1977), 111-129.
    • (1977) Studia Math. , vol.60 , Issue.2 , pp. 111-129
    • Jones, R.L.1
  • 47
    • 0039518933 scopus 로고
    • On the speed of convergence in the ergodic theorem
    • U. Krengel, "On the speed of convergence in the ergodic theorem", Monatsh. Math. 86:1 (1978), 3-6.
    • (1978) Monatsh. Math. , vol.86 , Issue.1 , pp. 3-6
    • Krengel, U.1
  • 48
    • 0039518933 scopus 로고
    • On the speed of convergence in the ergodic theorem
    • U. Krengel, "On the speed of convergence in the ergodic theorem", Monatsh. Math. 86:1 (1978), 3-6.
    • (1978) Monatsh. Math. , vol.86 , Issue.1 , pp. 3-6
    • Krengel, U.1
  • 49
    • 0000895088 scopus 로고
    • Large deviations of sums of independent random variables
    • S. V. Nagaev, "Large deviations of sums of independent random variables", Ann. Probab. 7 (1979), 745-789.
    • (1979) Ann. Probab. , vol.7 , pp. 745-789
    • Nagaev, S.V.1
  • 50
    • 84956085576 scopus 로고
    • Internal set theory: A new approach to nonstandard analysis
    • E. Nelson, "Internal set theory: a new approach to nonstandard analysis", Bull. Amer. Math. Soc. 83 (1977), 1165-1198.
    • (1977) Bull. Amer. Math. Soc. , vol.83 , pp. 1165-1198
    • Nelson, E.1
  • 51
    • 84956085576 scopus 로고
    • Internal set theory: A new approach to nonstandard analysis
    • E. Nelson, "Internal set theory: a new approach to nonstandard analysis", Bull. Amer. Math. Soc. 83 (1977), 1165-1198.
    • (1977) Bull. Amer. Math. Soc. , vol.83 , pp. 1165-1198
    • Nelson, E.1
  • 52
    • 0002286325 scopus 로고
    • Proof of the quasi-ergodic hypothesis
    • J. von Neumann, "Proof of the quasi-ergodic hypothesis", Proc. Nat. Acad. Sci. USA 18 (1932), 70-82.
    • (1932) Proc. Nat. Acad. Sci. USA , vol.18 , pp. 70-82
    • Von Neumann, J.1
  • 53
    • 0002286325 scopus 로고
    • Physical applications of the ergodic hypothesis
    • J. von Neumann, "Physical applications of the ergodic hypothesis", Proc. Nat. Acad. Sci. USA 18 (1932), 263-266.
    • (1932) Proc. Nat. Acad. Sci. USA , vol.18 , pp. 263-266
    • Von Neumann, J.1
  • 54
    • 0039518931 scopus 로고
    • Obtaining prescribed rates of convergence for the ergodic theorem
    • G. L. O'Brien, "Obtaining prescribed rates of convergence for the ergodic theorem", Canad. J. Math. 35 (1983), 1129-1146.
    • (1983) Canad. J. Math. , vol.35 , pp. 1129-1146
    • O'Brien, G.L.1
  • 55
    • 51249174505 scopus 로고
    • Convergence rates of the strong law for stationary mixing sequences
    • M. Peligrad, "Convergence rates of the strong law for stationary mixing sequences", Z. Wahrsch. Verw. Gebiete 70 (1985), 307-314.
    • (1985) Z. Wahrsch. Verw. Gebiete , vol.70 , pp. 307-314
    • Peligrad, M.1
  • 56
    • 0000951165 scopus 로고
    • On stable sequences of events
    • A. Renyi, "On stable sequences of events", Sankhyā. Ser. A 25 (1963), 293-302.
    • (1963) Sankhyā. Ser. A , vol.25 , pp. 293-302
    • Renyi, A.1
  • 57
    • 84968495062 scopus 로고
    • Sums of stationary random variables
    • E. A. Robinson, "Sums of stationary random variables", Proc. Amer. Math. Soc. 11 (1960), 77-79.
    • (1960) Proc. Amer. Math. Soc. , vol.11 , pp. 77-79
    • Robinson, E.A.1
  • 58
    • 0040704195 scopus 로고
    • n under moment restrictions
    • nunder moment restrictions", Ann. Math. Statist. 41 (1970), 1235-1248.
    • (1970) Ann. Math. Statist. , vol.41 , pp. 1235-1248
    • Serfling, R.J.1
  • 59
    • 38248999802 scopus 로고
    • Almost sure invariance principles for mixing sequences of random variables
    • Q.-M. Shao, "Almost sure invariance principles for mixing sequences of random variables", Stochastic Process. Appl. 48 (1993), 319-334
    • (1993) Stochastic Process. Appl. , vol.48 , pp. 319-334
    • Shao, Q.-M.1
  • 60
    • 0040110588 scopus 로고
    • Subadditive stochastic processes
    • M. D. Smeltzer, "Subadditive stochastic processes", Bull. Amer. Math. Soc. 83 (1977), 1054-1055.
    • (1977) Bull. Amer. Math. Soc. , vol.83 , pp. 1054-1055
    • Smeltzer, M.D.1
  • 61
    • 0001943778 scopus 로고
    • Sums of random variables with φ-mixing
    • S. A. Utev, "Sums of random variables with φ-mixing", Siberian Adv. Math. 1:3 (1991), 124-155.
    • (1991) Siberian Adv. Math. , vol.1 , Issue.3 , pp. 124-155
    • Utev, S.A.1
  • 62
    • 0037960194 scopus 로고
    • Über die gleichverteilung von Zahlen mod 1
    • H. Weyl, "Über die Gleichverteilung von Zahlen mod 1", Math. Ann. 77 (1916), 313-352.
    • (1916) Math. Ann. , vol.77 , pp. 313-352
    • Weyl, H.1
  • 63
    • 0038926348 scopus 로고
    • Some inequalities for ergodic power functions
    • T. Yoshimoto, "Some inequalities for ergodic power functions", Acta Math. Hungar. 36 (1980), 1-5.
    • (1980) Acta Math. Hungar. , vol.36 , pp. 1-5
    • Yoshimoto, T.1


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