The Law of the Iterated Logarithm for Random Dynamical System with Jumps and State-Dependent Jump Intensity

[1] D. Czapla, S.C. Hille, K. Horbacz, and H. Wojewódka-Sciazko, The law of the iterated logarithm for a piecewise deterministic Markov process assured by the properties of the Markov chain given by its post-jump locations, Stoch. Anal. Appl. 39 2021, no. 2, 357–379.10.1080/07362994.2020.1798252Search in Google Scholar

[2] D. Czapla, K. Horbacz, and H. Wojewódka-Ściążko, Ergodic properties of some piecewise-deterministic Markov process with application to a gene expression modelling, Stochastic Process. Appl. 130 2020, no. 5, 2851–2885.10.1016/j.spa.2019.08.006Search in Google Scholar

[3] D. Czapla, K. Horbacz, and H. Wojewódka-Ściążko, The Strassen invariance principle for certain non-stationary Markov-Feller chains, Asymptot. Anal. 121 2021, no. 1, 1–34.10.3233/ASY-191592Search in Google Scholar

[4] 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, 130–156.10.1080/14689367.2018.1485879Search in Google Scholar

[5] 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, 353–388.10.1111/j.2517-6161.1984.tb01308.xSearch in Google Scholar

[6] S.C. Hille, K. Horbacz, T. Szarek, and H.Wojewódka, Limit theorems for some Markov chains, J. Math. Anal. Appl. 443 2016, no. 1, 385–408.10.1016/j.jmaa.2016.05.022Search in Google Scholar

[7] A. Khintchine, Über einen Satz der Wahrscheinlichkeitsrechnung, Fund. Math. 6 1924, 9–20.10.4064/fm-6-1-9-20Search in Google Scholar

[8] A. Kolmogoroff, Über das Gesetz des iterierten Logarithmus, Math. Ann. 101 1929, 126–135.10.1007/BF01454828Search in Google Scholar

[9] T. Komorowski, C. Landim, and S. Olla, Fluctuations in Markov Processes. Time Symmetry and Martingale Approximation, Grundlehren der Mathematischen Wissenschaften, 345, Springer, Heidelberg, 2012.10.1007/978-3-642-29880-6_11Search in Google Scholar

[10] T. Komorowski and A. Walczuk, Central limit theorem for Markov processes with spectral gap in the Wasserstein metric, Stochastic Process. Appl. 122 2012, no. 5, 2155–2184.10.1016/j.spa.2012.03.006Search in Google Scholar

[11] J. Kubieniec, Random dynamical systems with jumps and with a function type intensity, Ann. Math. Sil. 30 2016, 63–87.10.1515/amsil-2016-0004Search in Google Scholar

[12] 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 Physics, 457, Springer, Berlin, 1995, pp. 235–255.10.1007/3-540-60188-0_58Search in Google Scholar

[13] 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

[14] T. Lipniacki, P. Paszek, A. Marciniak-Czochra, A.R. Brasier, and M. Kimmel, Transcriptional stochasticity in gene expression, J. Theoret. Biol. 238 2006, no. 2, 348–367.10.1016/j.jtbi.2005.05.03216039671Search in Google Scholar

[15] M.C. Mackey, M. Tyran-Kaminska, and R. Yvinec, Dynamic behavior of stochastic gene expression models in the presence of bursting, SIAM J. Appl. Math. 73 2013, no. 5, 1830–1852.10.1137/12090229XSearch in Google Scholar

[16] O. Zhao and M. Woodroofe, Law of the iterated logarithm for stationary processes, Ann. Probab. 36 2008, no. 1, 127–142.10.1214/009117907000000079Search in Google Scholar