Continuous Dependence in a Problem of Convergence of Random Iteration

• Autores: Rafal Kapica, Dawid Komorek
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 2, 2023
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • We show that an invariant measure of a Markov operator that is contracting in the Hutchison distance, acting on a space of Borel probability measures on a Polish space, depends continuously on the given operator. In addition, we establish an estimate for a distance between invariant measures. Some applications to the weak limit of iterates of random-valued functions (in particular, the so-called random affine maps, occuring, e.g., in perpetuities analysis) are also given.

• Referencias bibliográficas
  • 1. Aïssani, D., Kartashov, N.V.: Ergodicity and stability of Markov chains with respect to operator topology in the space of transition kernels....
  • 2. Alsmeyer, G., Iksanov, A., Rösler, U.: On distributional properties of perpetuities. J. Theor. Probab. 22, 666–682 (2009)
  • 3. Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics....
  • 4. Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)
  • 5. Baron, K., Kuczma, M.: Iteration of random-valued functions on the unit interval. Colloq. Math. 37, 263–269 (1977)
  • 6. Baron, K.: On the continuous dependence in a problem of convergence of iterates of random-valued functions. Grazer Math. Ber. 363, 1–6...
  • 7. Baron, K.: Remarks connected with the weak limit of iterates of some random-valued functions and iterative functional equations. Ann. Math....
  • 8. Baron, K.: Around the weak limit of iterates of some random-valued functions. Ann. Univ. Sci. Bp. Sect. Comput. 51, 31–37 (2020)
  • 9. Billingsley, P.: Probability and Measure. Wiley, New York (2012)
  • 10. Bogachev, V.I.: Measure Theory, vol. I, II. Springer, Berlin (2007)
  • 11. Czapla, D., Hille, S.C., Horbacz, K., Wojewódka-Sci¸ ´ a˙zko, H.: Continuous dependence of an invariant measure on the jump rate of a...
  • 12. Diamond, Ph.: A stochastic functional equation. Aequat. Math. 15, 225–233 (1977)
  • 13. Dudley, R.M.: Real Analysis and Probability. Cambridge Studies in Advanced Mathematics, vol. 74. Cambridge Univerity Press, Cambridge...
  • 14. Fortet, R., Mourier, E.: Convergence de la répartition empirique vers la répartition théorique. Ann. Sci. Ecole Norm. Sup. 70, 267–285...
  • 15. Goldie, C.M., Maller, R.A.: Stability of perpetuities. Ann. Probab. 28, 1195–1218 (2000)
  • 16. Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)
  • 17. Grinceviˇcjus, A.K.: On the continuity of the distribution of a sum of dependent variables connected with independent walks on lines....
  • 18. Hutchinson, J.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)
  • 19. Iksanov, A.: Renewal Theory for Perturbed Random Walks and Similar Processes, Probability and Its Applications. Birkhäuser, Basel (2016)
  • 20. Kapica, R.: The geometric rate of convergence of random iteration in the Hutchinson distance. Aequat. Math. 93, 149–160 (2019)
  • 21. Kartashov, N.V.: Strong Stable Markov Chains. TbiMC Scientific Publishers, VSP, Utrecht (1996)
  • 22. Kato, T.: Perturbation Theory for Linear Operators, 2nd corr. print. of the 2nd ed. Grundlehren der mathematischen Wissenschaften, vol....
  • 23. Kesten, H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248 (1973)
  • 24. Lasota, A., Mackey, M.C.: Chaos, Fractals and Noise. Stochastic Aspects of Dynamics. Applied Mathematical Sciences, vol. 97. Springer,...
  • 25. Lasota, A.: From fractals to stochastic differential equations. In: Lecture Notes in Physics, vol. 457, pp. 235–255 (1995)
  • 26. Luther, N.Y.: Locally compact spaces of measures. Proc. Am. Math. Soc. 25, 541–547 (1970)
  • 27. Nadler, S.B., Jr.: Sequences of contractions and fixed points. Pac. J. Math. 27, 579–585 (1968)
  • 28. Topsøe, F.: Compactness and tightness in a space of measures with the topology of weak convergence. Math. Scand. 34, 187–210 (1974)
  • 29. Vervaat, W.: On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl....
  • 30. Villani, C.: Optimal Transport. Old and New. Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences,...
  • 31. Zaharopol, R.: Invariant Probabilities of Transition Functions, Probability and Its Applications. Springer, Berlin (2014)