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Research Article

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

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Pages 357-379 | Received 11 Oct 2019, Accepted 15 Jul 2020, Published online: 03 Sep 2020

References

  • Khinchin, A. (1923). Über dyadische brüche (German). Math. Z. 18(1):109–116.
  • Kolmogorov, A. (1929). Über das gesetz des iterierten logarithmus. (German). Math. Ann. 101(1):126–135.
  • Hartman, P., Wintner, A. (1941). On the law of the iterated logarithm. Am. J. Math. 63(1):169–176. DOI: 10.2307/2371287.
  • Strassen, V. (1964). An invariance principle for the law of the iterated logarithm. Z Wahrscheinlichkeitstheorie Verw. Gebiete. 3(3):211–226. DOI: 10.1007/BF00534910.
  • Bingham, N. (1986). Variants on the law of the iterated logarithm. Lond. Math. Soc. 18(5):433–467. DOI: 10.1112/blms/18.5.433.
  • Czapla, D., Horbacz, K., Wojewódka-Ściążko, H. (2020a). The Strassen invariance principle for certain non-stationary Markov–Feller chains. ASY. 1–34. DOI: 10.3233/ASY-191592.
  • Davis, M. (1984). Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models. J. Roy. Statist. Soc. Ser. B. 46(3):353–376. DOI: 10.1111/j.2517-6161.1984.tb01308.x.
  • Mackey, M., Tyran-Kamińska, M., Yvinec, R. (2013). Dynamic behavior of stochastic gene expression models in the presence of bursting. SIAM J. Appl. Math. 73(5):1830–1852. DOI: 10.1137/12090229X.
  • Hille, S., Horbacz, K., Szarek, T. (2016). 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(2):171–217. DOI: 10.5802/ambp.360.
  • Riedler, M., Thieullen, M., Wainrib, G. (2012). Limit theorems for infinite-dimensional piecewise deterministic Markov processes. Applications to stochastic excitable membrane models. Electron. J. Probab. 17(0):1–48. DOI: 10.1214/EJP.v17-1946.
  • Alkurdi, T., Hille, S., Van Gaans, O. (2015). Persistence of stability for equilibria of map iterations in Banach spaces under small perturbations. Potent. Anal. 42(1):175–201. DOI: 10.1007/s11118-014-9429-2.
  • Benaïm, M., Le Borgne, S., Malrieu, F., Zitt, P.-A. (2012). Quantitative ergodicity for some switched dynamical systems. Electron. Commun. Probab. 17(0):14. DOI: 10.1214/ECP.v17-1932.
  • Benaïm, M., Le Borgne, S., Malrieu, F., Zitt, P.-A. (2015). Qualitative properties of certain piecewise deterministic Markov processes. Ann. Inst. H Poincaré Probab. Statist. 51(3):1040–1075. DOI: 10.1214/14-AIHP619.
  • Costa, O., Dufour, F. (2008). Stability and ergodicity of piecewise deterministic Markov processes. SIAM J. Control Optim. 47(2):1053–1077. DOI: 10.1137/060670109.
  • Dufour, F., Costa, O. (1999). Stability of piecewise-deterministic Markov processes. SIAM J. Control Optim. 37(5):1483–1502. DOI: 10.1137/S0363012997330890.
  • Czapla, D., Horbacz, K., Wojewódka-Ściążko, H. (2019). Ergodic properties of some piecewise-deterministic Markov process with application to gene expression modelling. Stoch. Process. Appl. 130(5):2851–2885. DOI: 10.1016/j.spa.2019.08.006.
  • Wojewódka, H. (2013). Exponential rate of convergence for some Markov operators. Statist. Probab. Lett. 83(10):2337–2347. DOI: 10.1016/j.spl.2013.05.035.
  • Zhao, O., Woodroofe, M. (2008). Law of the iterated logarithm for stationary processes. Ann. Probab. 36(1):127–142. DOI: 10.1214/009117907000000079.
  • Bołt, W., Majewski, A., Szarek, T. (2012). An invariance principle for the law of the iterated logarithm for some Markov chains. Stud. Math. 212(1):41–53. DOI: 10.4064/sm212-1-3.
  • Czapla, D., Horbacz, K., Wojewódka-Ściążko, H. (2020b). A useful version of the central limit theorem for a general class of Markov chains. J. Math. Anal. Appl. 484(1):123725.
  • Komorowski, T., Landim, C., Olla, S. (2012). Fluctuations in Markov Processes. Time Symmetry and Martingale Approximation. Berlin: Springer.
  • Heyde, C. C., Scott, D. J. (1973). Invariance principles for the law of the iterated logarithm for martingales and processes with stationary increments. Ann. Probab. 1(3):428–436. DOI: 10.1214/aop/1176996937.
  • Hairer, M. (2002). Exponential mixing properties of stochastic PDEs through asymptotic coupling. Probab. Theory Relat. Fields. 124(3):345–380. DOI: 10.1007/s004400200216.
  • Hillen, T., Othmer, H. (2000). The diffusion limit of transport equations derived from velocity-jump processes. SIAM J. Appl. Math. 61(3):751–775. DOI: 10.1137/S0036139999358167.
  • Othmer, H., Dunbar, S., Alt, W. (1988). Models of dispersal in biological systems. J. Math. Biol. 26(3):263–298. DOI: 10.1007/BF00277392.
  • Perthame, B. (2007). Transport Equations in Biology. Basel: Birkhäuser Basel.
  • Lasota, A. (1995). From fractals to stochastic differential equations. In: Chaos: The Interplay Between Stochastic and Deterministic Behaviour. Lecture Notes in Physics, No. 457. Berlin: Springer, pp. 235–255.
  • Dudley, R. (1976). Probabilities and Metrics. Convergence of Laws on Metric Spaces, with a View to Statistical Testing. Lecture Notes Series, No. 45. Aarhus: Matematisk Institut, Aarhus Universitet.
  • Revuz, D. (1974). Markov Chains. Amsterdam: North-Holland Elsevier.
  • Czapla, D., Kubieniec, J. (2019). Exponential ergodicity of some Markov dynamical systems with application to a Poisson driven stochastic differential equation. Dyn. Syst. 34(1):130–156. DOI: 10.1080/14689367.2018.1485879.
  • Kapica, R., Ślęczka, M. (2020). Random iterations with place dependent probabilities. Probab. Math. Statist. DOI: 10.37190/0208-4147.40.1.8.

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