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

Full-text PDF

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]**P. Billingsley,*Convergence of Probability Measures*, J.Wiley, New York, 1968. MR**38:1718****[Bi2]**P. Billingsley,*Probability and Measure*, J.Wiley, New York, 1986. MR**87f:60001****[Bo]**R. Bowen,*Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms*, Lect. Notes in Math. 470, Springer-Verlag, Berlin, 1975. MR**56:1364****[BG1]**T. Bogenschutz and V. M. Gundlach,*Symbolic dynamics for expanding random dynamical systems*, Random & Comp. Dyn.**1**(1992), 219-227. MR**93j:58042****[BG2]**T. Bogenschutz and V. M. Gundlach,*Ruelle's transfer operator for random subshifts of finite type*, Ergod. Th. & Dynam. Sys.**15**(1995), 413-447. MR**96m:58133****[Co]**R. Cogburn,*On the central limit theorem for Markov chains in random environments*, Ann. Probab.**19**(1991), 587-604. MR**92h:60029****[CFS]**I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai,*Ergodic Theory*, Springer-Verlag, Berlin, 1982. MR**87f:28019****[De]**M. Denker,*The central limit theorem for dynamical systems*, Dynamical Systems and Ergodic Theory, Banach Center Publ., vol. 23, PWN-Polish Sci. Publ., Warszawa, 1989, pp. 33-62. MR**92d:28007****[Do]**P. Doukhan,*Mixing: Properties and Examples*, Lecture Notes in Stat. 85, Springer-Verlag, New York, 1994.MR**96b:60090****[DP]**M. Denker and W. Philipp,*Approximation by Brownian motion for Gibbs measures and flows under a function*, Ergod. Th. & Dynam. Sys.**4**(1984), 541-552. MR**86g:28025****[Go]**M. I. Gordin,*The central limit theorem for stationary processes*, Soviet Math. Dokl.**10**(1969), 1174-1176. MR**40:5012****[Gu]**V. M. Gundlach,*Thermodynamic formalism for random subshifts of finite type*, Preprint, 1996.**[Ha]**P. Hartman,*Ordinary Differential Equations*, J.Wiley, New York, 1964. MR**30:1270****[HH]**P. Hall and C. C. Heyde,*Martingale Limit Theory and its Applications*, Acad. Press, New York, 1980. MR**83a:60001****[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. Probab.**1**(1973), 428-436. MR**50:5886****[IL]**I.A.Ibragimov and Y.V.Linnik,*Independent and Stationary Sequences of Random Variables*, Wolters-Nordhoff, Groningen, 1971. MR**48:1287****[Kh]**R.Z.Khasminskii,*On stochastic processes defined by differential equations with a small parameter*, Theory Probab. and its Appl.**11**(1966), 211-228. MR**34:3636****[Ki1]**Y. Kifer,*Equilibrium states for random expanding transformations*, Random & Comp. Dyn.**1**(1992), 1-31. MR**93i:58075****[Ki2]**Y. Kifer,*Limit theorems in averaging for dynamical systems*, Ergod. Th. & Dynam. Sys.**15**(1995), 1143-1172. MR**97e:58087****[Ki3]**Y. Kifer,*Perron-Frobenius theorem, large deviations, and random perturbations in random environments*, Math. Zeitschrift**222**(1996), 677-698. MR**97f:60131****[Ki4]**Y. Kifer,*Fractal dimensions and random transformations*, Trans. Amer. Math. Soc.**348**(1996), 2003-2038. MR**96i:28009****[KK]**K. Khanin and Y. Kifer,*Thermodynamic formalism for random transformations and statistical mechanics*, in: Sinai's Moscow Seminar on Dynamical Systems (L.A.Bunimovich, B.M.Gurevich, Ya.B.Pesin, eds.), American Mathematical Society Translations (2)**171**(1995), 107-140. MR**96j:58136****[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]**S. Orey,*Markov chains with stochastically stationary transition probabilities*, Ann. Probab.**19**(1991), 907-928. MR**92i:60065****[PS]**W. Philipp and W. F. Stout,*Almost sure invariance principles for sums of weakly dependent random variables*, Memoirs of Amer. Math. Soc. 161, Amer. Math. Soc., Providence, 1975. MR**55:6570****[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]**T. Seppäläinen,*Large deviations for Markov chains with random transitions*, Ann. Probab.**22**(1994), 713-748. MR**95j:60044****[Sh]**A.N.Shiryaev,*Probability*, Springer-Verlag, New York, 1984. MR**85a:60007****[VR]**V.A.Volkonskii and Yu.A.Rozanov,*Some limit theorems for random functions I*, Th. Probab. Appl.**4**(1959), 178-197. MR**22:12586**; MR**21:4477**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
60F05,
58F15,
60J99,
60J05

Retrieve articles in all journals with MSC (1991): 60F05, 58F15, 60J99, 60J05

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