Limit theorems for random transformations and processes in random environments

Author:
Yuri Kifer

Journal:
Trans. Amer. Math. Soc. **350** (1998), 1481-1518

MSC (1991):
Primary 60F05; Secondary 58F15, 60J99, 60J05

DOI:
https://doi.org/10.1090/S0002-9947-98-02068-6

MathSciNet review:
1451607

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Abstract | References | Similar Articles | Additional Information

Abstract: I derive general relativized central limit theorems and laws of iterated logarithm for random transformations both via certain mixing assumptions and via the martingale differences approach. The results are applied to Markov chains in random environments, random subshifts of finite type, and random expanding in average transformations where I show that the conditions of the general theorems are satisfied and so the corresponding (fiberwise) central limit theorems and laws of iterated logarithm hold true in these cases. I consider also a continuous time version of such limit theorems for random suspensions which are continuous time random dynamical systems.

**[Bi1]**Patrick Billingsley,*Convergence of probability measures*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0233396****[Bi2]**Patrick Billingsley,*Probability and measure*, 2nd ed., Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. MR**830424****[Bo]**Rufus Bowen,*Equilibrium states and the ergodic theory of Anosov diffeomorphisms*, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR**0442989****[BG1]**Thomas Bogenschütz and Volker Matthias Gundlach,*Symbolic dynamics for expanding random dynamical systems*, Random Comput. Dynam.**1**(1992/93), no. 2, 219–227. MR**1186374****[BG2]**Thomas Bogenschütz and Volker Mathias Gundlach,*Ruelle’s transfer operator for random subshifts of finite type*, Ergodic Theory Dynam. Systems**15**(1995), no. 3, 413–447. MR**1336700**, https://doi.org/10.1017/S0143385700008464**[Co]**Robert Cogburn,*On the central limit theorem for Markov chains in random environments*, Ann. Probab.**19**(1991), no. 2, 587–604. MR**1106277****[CFS]**I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ,*Ergodic theory*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR**832433****[De]**Manfred Denker,*The central limit theorem for dynamical systems*, Dynamical systems and ergodic theory (Warsaw, 1986) Banach Center Publ., vol. 23, PWN, Warsaw, 1989, pp. 33–62. MR**1102700****[Do]**Paul Doukhan,*Mixing*, Lecture Notes in Statistics, vol. 85, Springer-Verlag, New York, 1994. Properties and examples. MR**1312160****[DP]**Manfred Denker and Walter Philipp,*Approximation by Brownian motion for Gibbs measures and flows under a function*, Ergodic Theory Dynam. Systems**4**(1984), no. 4, 541–552. MR**779712**, https://doi.org/10.1017/S0143385700002637**[Go]**M. I. Gordin,*The central limit theorem for stationary processes*, Dokl. Akad. Nauk SSSR**188**(1969), 739–741 (Russian). MR**0251785****[Gu]**V. M. Gundlach,*Thermodynamic formalism for random subshifts of finite type*, Preprint, 1996.**[Ha]**Philip Hartman,*Ordinary differential equations*, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR**0171038****[HH]**P. Hall and C. C. Heyde,*Martingale limit theory and its application*, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. Probability and Mathematical Statistics. MR**624435****[HS]**C. C. Heyde and D. J. Scott,*Invariance principles for the law of the iterated logarithm for martingales and processes with stationary increments*, Ann. Probability**1**(1973), 428–436. MR**353403**, https://doi.org/10.1214/aop/1176996937**[IL]**I. A. Ibragimov and Yu. V. Linnik,*Independent and stationary sequences of random variables*, Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov; Translation from the Russian edited by J. F. C. Kingman. MR**0322926****[Kh]**R. Z. Has′minskiĭ,*Stochastic processes defined by differential equations with a small parameter*, Teor. Verojatnost. i Primenen**11**(1966), 240–259 (Russian, with English summary). MR**0203788****[Ki1]**Wen Xiu Ma,*A hierarchy of Liouville integrable finite-dimensional Hamiltonian systems*, Appl. Math. Mech.**13**(1992), no. 4, 349–357 (Chinese, with English summary); English transl., Appl. Math. Mech. (English Ed.)**13**(1992), no. 4, 369–377. MR**1169984**, https://doi.org/10.1007/BF02451423**[Ki2]**Manuel De León and David Martín de Diego,*Almost product structures in mechanics*, Differential geometry and applications (Brno, 1995) Masaryk Univ., Brno, 1996, pp. 539–548. MR**1406373****[Ki3]**Yuri Kifer,*Perron-Frobenius theorem, large deviations, and random perturbations in random environments*, Math. Z.**222**(1996), no. 4, 677–698. MR**1406273**, https://doi.org/10.1007/PL00004551**[Ki4]**Yuri Kifer,*Fractal dimensions and random transformations*, Trans. Amer. Math. Soc.**348**(1996), no. 5, 2003–2038. MR**1348865**, https://doi.org/10.1090/S0002-9947-96-01608-X**[KK]**K. Khanin and Y. Kifer,*Thermodynamic formalism for random transformations and statistical mechanics*, Sinaĭ’s Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, vol. 171, Amer. Math. Soc., Providence, RI, 1996, pp. 107–140. MR**1359097**, https://doi.org/10.1090/trans2/171/10**[Li]**C. Liverani,*Central limit theorem for deterministic systems*, in: Proc. Int. Congr. in Dynam. Sys. (F.Ledrappier, J.Lewowicz, and S.Newhouse, eds.), Pitman Research Notes in Mathematics, Longman, Harlow, 1996.**[Or]**Steven Orey,*Markov chains with stochastically stationary transition probabilities*, Ann. Probab.**19**(1991), no. 3, 907–928. MR**1112400****[PS]**Walter Philipp and William Stout,*Almost sure invariance principles for partial sums of weakly dependent random variables*, Mem. Amer. Math. Soc. 2**issue 2, 161**(1975), iv+140. MR**0433597**, https://doi.org/10.1090/memo/0161**[Ru]**B.-Z. Rubshtein,*A central limit theorem for conditional distributions*, in: Convergence in Ergodic Theory and Probability (Bergelson, March, Rosenblatt, eds.), Walter de Gruyter, Berlin, 1996. CMP**97:02****[Se]**Timo Seppäläinen,*Large deviations for Markov chains with random transitions*, Ann. Probab.**22**(1994), no. 2, 713–748. MR**1288129****[Sh]**A. N. Shiryayev,*Probability*, Graduate Texts in Mathematics, vol. 95, Springer-Verlag, New York, 1984. Translated from the Russian by R. P. Boas. MR**737192****[VR]**V. A. Volkonskiĭ and Yu. A. Rozanov,*Some limit theorems for random functions. I*, Theor. Probability Appl.**4**(1959), 178–197. MR**121856**, https://doi.org/10.1137/1104015

V. A. Volkonskiĭ and Yu. A. Rozanov,*Some limit theorems for random functions. I*, Teor. Veroyatnost. i Primenen**4**(1959), 186–207 (Russian, with English summary). MR**0105741**

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Additional Information

**Yuri Kifer**

Affiliation:
Institute of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel

Email:
kifer@math.huji.ac.il

DOI:
https://doi.org/10.1090/S0002-9947-98-02068-6

Keywords:
Central limit theorem,
random transformations,
random environment

Received by editor(s):
July 16, 1996

Additional Notes:
Supported by the US-Israel Binational Science Foundation

Article copyright:
© Copyright 1998
American Mathematical Society