Central Limit Theorem for Random Dynamical System with Jumps and State-Dependent Jump Intensity

Kubieniec, Joanna

doi:10.2478/amsil-2025-0004

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.

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.
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
D. Czapla and J. Kubieniec, Exponential ergodicity of some Markov dynamical systems with application to a Poisson-driven stochastic diflerential equation, Dyn. Syst. 34 (2019), no. 1, 130–156.
Search in Google Scholar Back to article
M.H.A. Davis, Piecewise-deterministic Markov processes: a general class of nondiflusion stochastic models, J. Roy. Statist. Soc. Ser. B 46 (1984), no. 3, 353–388.
Search in Google Scholar Back to article
F. Dufour and O.L.V. Costa, Stability of piecewise-deterministic Markov processes, SIAM J. Control Optim. 37 (1999), no. 5, 1483–1502.
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
K. Ito and F. Kappel, Evolution Equations and Approximations, Ser. Adv. Math. Appl. Sci., 61, World Scientific Publishing Co., Inc., River Edge, NJ, 2002.
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
A. Lasota, From fractals to stochastic diflerential 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.
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
T. Szarek, Invariant measures for Markov operators with application to function systems, Studia Math. 154 (2003), no. 3, 207–222.
Search in Google Scholar Back to article
H. Wojewódka, Exponential rate of convergence for some Markov operators, Statist. Probab. Lett. 83 (2013), no. 10, 2337–2347.
Search in Google Scholar Back to article

Central Limit Theorem for Random Dynamical System with Jumps and State-Dependent Jump Intensity

Abstract

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