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

• Autorzy: Joanna Kubieniec
• Języki publikacji: EN

Abstrakty

EN

In this paper our considerations are focused on some Markov chain associated with certain piecewise-deterministic Markov process with a statedependent jump intensity for which the exponential ergodicity was obtained in [4]. Using the results from [3] we show that the law of iterated logarithm holds for such a model.(original abstract)

Słowa kluczowe

EN

Mathematics; Mathematical economics; Markov chain; Calculus of probability

PL

Matematyka; Ekonomia matematyczna; Łańcuch Markowa; Rachunek prawdopodobieństwa

• Czasopismo: Annales Mathematicae Silesianae
• Rocznik: 2021
• Tom: 35 (2)
• Strony: 236--249

Twórcy

autor

Joanna Kubieniec

• University of Silesia in Katowice, Poland

Bibliografia

• Czapla D., Hille S.C., Horbacz K., and Wojewódka-Sciążko H., 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.
• Czapla D., Horbacz K., and Wojewódka-Sciążko H., Ergodic properties of some piecewise-deterministic Markov process with application to a gene expression modelling, Stochastic Process. Appl. 130 (2020), no. 5, 2851-2885.
• Czapla D., Horbacz K., and Wojewódka-Sciążko H., The Strassen invariance principle for certain non-stationary Markov-Feller chains, Asymptot. Anal. 121 (2021), no. 1, 1-34.
• Czapla D. and Kubieniec J., Exponential ergodicity of some Markov dynamical systems with application to a Poisson-driven stochastic differential equation, Dyn. Syst. 34 (2019), no. 1, 130-156.
• Davis M.H.A., Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B 46 (1984), no. 3, 353-388.
• Hille S.C., Horbacz K., Szarek T., and Wojewódka H., Limit theorems for some Markov chains, J. Math. Anal. Appl. 443 (2016), no. 1, 385-408.
• Khintchine A., Über einen Satz der Wahrscheinlichkeitsrechnung, Fund. Math. 6 (1924), 9-20.
• Kolmogoroff A., Über das Gesetz des iterierten Logarithmus, Math. Ann. 101 (1929), 126-135.
• Komorowski T., Landim C., and Olla S., Fluctuations in Markov Processes. Time Symmetry and Martingale Approximation, Grundlehren der Mathematischen Wissenschaften, 345, Springer, Heidelberg, 2012.
• Komorowski T. and Walczuk A., Central limit theorem for Markov processes with spectral gap in the Wasserstein metric, Stochastic Process. Appl. 122 (2012), no. 5, 2155-2184.
• Kubieniec J., Random dynamical systems with jumps and with a function type intensity, Ann. Math. Sil. 30 (2016), 63-87.
• Lasota A., 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.
• Lasota A. and Yorke J.A., Lower bound technique for Markov operators and iterated function systems, Random Comput. Dynam. 2 (1994), no. 1, 41-77.
• Lipniacki T., Paszek P., Marciniak-Czochra A., Brasier A.R., and Kimmel M., Transcriptional stochasticity in gene expression, J. Theoret. Biol. 238 (2006), no. 2, 348-367.
• Mackey M.C., Tyran-Kamińska M., and Yvinec R., Dynamic behavior of stochastic gene expression models in the presence of bursting, SIAM J. Appl. Math. 73 (2013), no. 5, 1830-1852.
• Zhao O. and Woodroofe M., Law of the iterated logarithm for stationary processes, Ann. Probab. 36 (2008), no. 1, 127-142.

Identyfikatory

DOI

10.2478/amsil-2021-0011