Central limit theorem for random dynamical system with jumps and state-dependent jump intensity

Joanna Kubieniec

Published: 2025-03-09

2024

Central limit theorem for random dynamical system with jumps and state-dependent jump intensity

Joanna Kubieniec

Abstract

In this work we focus on a dynamical system with jumps, where the intensity of the jumps depends on the system’s state. By verifying the assumptions of the theorem from [4], we show that our model satisfies the central limit theorem.

Keywords:

Markov operator , Markov chain , central limit theorem

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Kubieniec, J. (2025). Central limit theorem for random dynamical system with jumps and state-dependent jump intensity. Annales Mathematicae Silesianae. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/18558

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References

T. Alkurdi, S. Hille, and O. van Gaans, Persistence of stability for equilibria of map iterations in Banach spaces under small random perturbations, Potential Anal. 42 (2015), no. 1, 175–201.
Google Scholar

M. Benaïm, S. Le Borgne, F. Malrieu, and P.-A. Zitt, Quantitative ergodicity for some switched dynamical systems, Electron. Commun. Probab. 17 (2012), no. 56, 14 pp.
Google Scholar

O.L.V. Costa and F. Dufour, Stability and ergodicity of piecewise deterministic Markov processes, SIAM J. Control Optim. 47 (2008), no. 2, 1053–1077.
Google Scholar

D. Czapla, K. Horbacz, and H. Wojewódka-Ściążko, A useful version of the central limit theorem for a general class of Markov chains, J. Math. Anal. Appl. 484 (2020), no. 1, 123725, 22 pp.
Google Scholar

D. Czapla, K. Horbacz, and H. Wojewódka-Ściążko, Ergodic properties of some piecewise-deterministic Markov process with application to gene expression modelling, Stochastic Process. Appl. 130 (2020), no. 5, 2851–2885.
Google Scholar

D. Czapla and J. Kubieniec, Exponential ergodicity of some Markov dynamical systems with application to a Poisson-driven stochastic differential equation, Dyn. Syst. 34 (2019), no. 1, 130–156.
Google Scholar

M.H.A. Davis, Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B 46 (1984), no. 3, 353–388.
Google Scholar

F. Dufour and O.L.V. Costa, Stability of piecewise-deterministic Markov processes, SIAM J. Control Optim. 37 (1999), no. 5, 1483–1502.
Google Scholar

S. Hille, K. Horbacz, and T. Szarek, Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene, Ann. Math. Blaise Pascal 23 (2016), no. 2, 171–217.
Google Scholar

K. Ito and F. Kappel, Evolution Equations and Approximations, Ser. Adv. Math. Appl. Sci., 61, World Scientific Publishing Co., Inc., River Edge, NJ, 2002.
Google Scholar

J. Kubieniec, The law of the iterated logarithm for random dynamical system with jumps and state-dependent jump intensity, Ann. Math. Sil. 35 (2021), no. 2, 236–249.
Google Scholar

A. Lasota, From fractals to stochastic differential equations, in: P. Garbaczewski et al. (eds.), Chaos - The Interplay Between Stochastic and Deterministic Behaviour, Lecture Notes in Phys., 457, Springer-Verlag, Berlin, 1995, pp. 235–255.
Google Scholar

A. Lasota and J.A. Yorke, Lower bound technique for Markov operators and iterated function systems, Random Comput. Dynam. 2 (1994), no. 1, 41–77.
Google Scholar

M.C. Mackey, M. Tyran-Kamińska, and R. Yvinec, Dynamic behavior of stochastic gene expression models in the presence of bursting, SIAM J. Appl. Math. 73 (2013), no. 5, 1830–1852.
Google Scholar

T. Szarek, Invariant measures for Markov operators with application to function systems, Studia Math. 154 (2003), no. 3, 207–222.
Google Scholar

H. Wojewódka, Exponential rate of convergence for some Markov operators, Statist. Probab. Lett. 83 (2013), no. 10, 2337–2347.
Google Scholar

2024
Published: 2024-01-18

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

Licence CC

This work is licensed under a Creative Commons Attribution 4.0 International License.

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