On the central limit theorem for homogeneous discrete-time nonlinear markov chains
- Authors: Shchegolev A.A.1
-
Affiliations:
- HSE University
- Issue: No 102 (2023)
- Pages: 5-14
- Section: Systems analysis
- URL: https://journals.rcsi.science/1819-2440/article/view/363789
- DOI: https://doi.org/10.25728/ubs.2023.102.1
- ID: 363789
Cite item
Full Text
Abstract
The class of nonlinear Markov processes is characterized by the dependence of the current state of the process on its current distribution in addition to the dependence on the previous state. Due to this feature, these processes are characterized by complex limit behavior and ergodic properties, for which the usual criteria for Markov processes are not sufficient. Being a subclass of nonlinear Markov processes, nonlinear Markov chains have inherited these features. In this paper, conditions for the fulfillment of the central limit theorem for homogeneous nonlinear Markov chains in discrete time and with a discrete finite state space are studied. Also, a brief review of known results on the ergodic properties of nonlinear Markov chains is included. The obtained result complements the existing results in this area and may be useful for further applications in statistics.
About the authors
Aleksandr Alekseevich Shchegolev
HSE University
Author for correspondence.
Email: ashchegolev@hse.ru
Moscow
References
- Бутковский О. А. Об эргодических свойствах нелинейных марковских цепей и стохастических уравнений Макина–Власова // Теория вероятностей и ее применения. – 2013. – Т. 58, №4. – С. 782–794.
- Власов А. А. О вибрационных свойствах электронного газа // Журнал экспериментальной и теоретической физики. – 1938. – Т. 8, №3. – С. 291–318.
- Ибрагимов И. А., Линник Ю. В. Независимые и стационарно связанные величины. – М.: Изд-во «Наука», Гл. ред. физ.-мат. лит-ры, 1965.
- Щеголев А. А. Об оценках скорости сходимости однородных нелинейных цепей Маркова в дискретном времени // Управление большими системами. – 2021. – Вып. 90. – С. 36–48.
- Kolokoltsov V. N. Nonlinear Markov processes and kinetic equations // Cambridge Tracts in Mathematics. – Cambridge: Cambridge University Press, 2010. – Vol. 182.
- McKean H. P. A class of Markov processes associated with nonlinear parabolic equations // Proc. of the National Academy of Sciences of the United States of America. – 1966. – Vol. 56. – P. 1907–1911.
- Muzychka S. A., Vaninsky K. L. A class of nonlinear random walks related to the Ornstein–Uhlenbeck process // Markov Process. Related Fields. – 2011. – Vol. 17, No. 2. – P. 277–304.
- Neumann B. Nonlinear Markov chains with finite state space: Invariant distributions and long-term behaviour // Journal of Applied Probability. – 2022. – P. 1–15.
- Saburov M. Ergodicity of nonlinear Markov operators on the finite dimensional space // Nonlinear Analysis: Theory, Methods & Applications. – 2015. – Vol. 143. – P. 105–119.
- Shchegolev A. A. A new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chains // Random Operators and Stochastic Equations. – 2022. – Vol. 30, No. 3. – P. 205–213.
- Sznitman A.-S. Topics in propagation of chaos // École d’Été de Probabilités de Saint-Flour XIX—1989. – Lecture Notes in Mathematics. – 1991. – Vol. 1464. – P. 165–251.
Supplementary files
