Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-26T12:01:57.718Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for stochastically perturbed dynamical systems

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Krzysztof Łoskot  and
Ryszard Rudnicki
Krzysztof Łoskot*
Affiliation:
Silesian University, Katowice
Ryszard Rudnicki*
Affiliation:
Polish Academy of Sciences, Katowice
*
∗Postal address: Institute of Mathematics, Silesian University, 40–007 Katowice, Poland.
∗∗Postal adress: Institute of Mathematics, Polish Academy of Sciences, Staromiejska 8/6, 40–013 Katowice, Poland.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a discrete-time stochastically perturbed dynamical system on the Polish space given by the recurrence formula Xn = S(Xn–1, Yn), where Yn are i.i.d. random elements. We prove the existence of unique stationary measure and versions of classical limit theorems for the process (Xn).

Keywords

INVARIANT MEASURELIMIT THEOREMS

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 2 , June 1995 , pp. 459 - 469
Copyright
Copyright © Applied Probability Trust 1995 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported by the State Committee for Scientific Research Grant No. 2 P301 026 05.

References

[1] Barnsley, M. F. (1988) Fractals Everywhere. Academic Press, New York.Google Scholar
[2] Barnsley, M. F. and Demko, S. G. (1985) Iterated function systems and the global construction of fractals. Proc. R. Soc. London A399, 243–275.Google Scholar
[3] Barnsley, M. F., Demko, S. G., Elton, J. H. and Geronimo, J. S. (1988) Invariant measures arising from iterated function systems with place dependent probabilities. Ann. Inst. H. Poincaré 24, 367–394.Google Scholar
[4] Barnsley, M. F., Ervin, V., Hardin, D. and Lancaster, J. (1986) Solutions of an inverse problem for fractals and other sets. Proc. Natl. Acad. Sci. U.S.A. 83, 1975–1977.CrossRefGoogle ScholarPubMed
[5] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[6] Boyarsky, A. (1980) Randomness implies order. J. Math. Anal. Appl. 76, 483–497.CrossRefGoogle Scholar
[7] Denker, M. (1989) The central limit theorem for dynamical systems. Dynamical System and Ergodic Theory, Banach Center Publications 23, pp. 33–63. PWN, Warszawa.Google Scholar
[8] Elton, J. H. and Piccioni, M. (1992) Iterated function systems arising from recursive estimation problems. Prob. Theory Rel. Fields 91, 103–114.CrossRefGoogle Scholar
[9] Horbacz, K. (1993) Dynamical systems with multiplicative perturbations: the strong convergence of measures. Ann. Pol. Math. 58, 85–93.CrossRefGoogle Scholar
[10] Ibragimov, I. A. (1975) A note on the central limit theorem for dependent random variables. Theory Prob. Appl. 20, 135–141.CrossRefGoogle Scholar
[11] Lasota, A. and Mackey, M. C. (1985) Probabilistic Properties of Deterministic Systems. Cambridge University Press.CrossRefGoogle Scholar
[12] Lasota, A. and Mackey, M. C. (1987) Noise and statistical periodicity. Physica D28, 143–154.Google Scholar
[13] Lasota, A. and Mackey, M. C. (1989) Stochastic perturbation of dynamical systems: the weak convergence of measures. J. Math. Anal. Appl. 138, 232–248.CrossRefGoogle Scholar
[14] Lasota, A. and Yorke, J. A. (1994) Lower bound technique for Markov operators and iteratred function systems. Random and Computational Dynamics 2, 41–77.Google Scholar
[15] Letac, G. (1986) A contraction principle for certain Markov chains and its applications. In Random Matrices and Their Applications, ed. Cohen, , Kesten, and Newman, , Contemp. Math. 50, pp. 263–273. Amer. Math. Soc. Providence, RI.CrossRefGoogle Scholar
[16] Shirjaev, A. N. (1989) Probability. Nauka, Moscow (in Russian).Google Scholar