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## On convergence rate bounds for a class of nonlinear Markov chains

Shchegolev A., Веретенников А. Ю.

In press

A new approach is developed for evaluating the convergence rate for nonlinear

Markov chains (MC) based on the recently developed spectral radius technique of

Markovian coupling for linear MC and the idea of small nonlinear perturbations of

linear MC. The method further enhances recent advances in the problem of convergence

for such models.

Language:
English

Keywords: convergence ratenonlinear Markov chainsuniform ergodicityMarkovian couplingspectral radius, extreme values

Publication based on the results of:

Shchegolev A., , in : Сборник материалов V-й Международной конференции по стохастическим методам: The 5th International Conference on Stochastic Methods (ICSM5). 23-27 November 2020, Russia, Moscow. : M. : RUDN, 2020. P. 191-196.

Added: October 30, 2021

Shchegolev A., Управление большими системами: сборник трудов 2021 № 90 С. 36-48

The paper studies an improved estimate for the rate of convergence for nonlinear homogeneous discrete-time Markov chains. These processes are nonlinear in terms of the distribution law. Hence, the transition kernels are dependent on the current probability distributions of the process apart from being dependent on the current state. Such processes often act as limits ...

Added: April 21, 2021

Piterbarg V., Щербакова Ю. А., Теория вероятностей и ее применения 2022 Т. 67 № 1 С. 57-80

We propose a sequence of accompanying laws in the B. V. Gnedenko limit theorem
for maxima of independent random variables with distributions lying in the Gumbel max domain
of attraction. We show that this sequence provides a power-law convergence rate, whereas the
Gumbel distribution provides only the logarithmic rate. As examples, we consider in detail the classes
of Weibull ...

Added: October 27, 2022

Aleksandr A. Shchegolev, Random Operators and Stochastic Equations 2022 Vol. 30 No. 3 P. 205-213

In the paper, we study a new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chains based on the Markov-Dobrushin condition. This result generalizes the convergence estimates for any positive number of transition steps. An example of a class such a process provided indicates that such types of estimates considering several transition steps may ...

Added: October 30, 2021

Shchegolev A., / Cornell University. Серия math "arxiv.org". 2021.

In the paper, we study a new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chains based on the Markov-Dobrushin condition. This result generalizes the convergence estimates for any positive number of transition steps. An example of a class such a process provided indicates that such types of estimates considering several transition steps may ...

Added: October 22, 2021

Vedenin A., Remizov I., / Cornell University. Series math "arxiv.org". 2020.

Abstract. The method of Chernoff approximation was discovered by Paul Chernoff in 1968 and now is a powerful and flexible tool of contemporary functional analysis. This method is different from grid-based approach and helps to solve numerically the Cauchy problem for evolution equations, e.g., for heat equation and for more general parabolic second-order partial differential equations ...

Added: December 14, 2020

Veretennikov A., / Cornell University. Series cond-mat "arxiv.org". 2014.

Polynomial convergence rate to stationarity is shown for extended Erlang -- Sevastyanov's model. ...

Added: December 16, 2014

Veretennikov A., Zverkina G., , in : DISTRIBUTED COMPUTER AND COMMUNICATION NETWORKS: CONTROL, COMPUTATION, COMMUNICATIONS (DCCN-2015), proceedings of the eighteenth international scientific conference. : M. : -, 2015.

A computable estimate of the readiness coefficient to its stationary value for a standard binary-state system is established in the case where working time and repair time distributions have heavy tails. ...

Added: October 27, 2015

Rodomanov A., Kropotov D., SIAM Journal on Optimization 2020 Vol. 30 No. 3 P. 1878-1904

We analyze the coordinate descent method with a new coordinate selection strategy, called volume sampling. This strategy prescribes selecting subsets of variables of certain size proportionally to the determinants of principal submatrices of the matrix, which bounds the curvature of the objective function. In the particular case when the size of the subsets equals one, ...

Added: July 29, 2020

Shchegolev A., Управление большими системами: сборник трудов 2023 № 102 С. 5-14

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 ...

Added: June 12, 2023

Rodomanov A., Kropotov D., , in : Proceedings of Machine Learning Research. Proceedings of the International Conference on Machine Learning (ICML 2016). Vol. 48.: NY : [б.и.], 2016. P. 2597-2605.

We consider the problem of minimizing the strongly convex sum of a finite number of convex functions. Standard algorithms for solving this problem in the class of incremental/stochastic methods have at most a linear convergence rate. We propose a new incremental method whose convergence rate is superlinear – the Newton-type incremental method (NIM). The idea ...

Added: December 10, 2018

Tkachenko A., Afanasyeva L. G., Journal of Mathematical Sciences 2016 Vol. 218 No. 2 P. 119-136

Convergence rates in total variation are established for some models of queueing theory and reliability theory. Analysis is based on renewal technique and asymptotic results for the renewal function. It is shown that convergence rate has an exponential asymptotics when distribution function of regeneration period satisfies Cramer's condition. Results concerning polynomial convergence are also obtained. ...

Added: October 14, 2015

Mammen E., Dunker V., Florens J. -. et al., Journal of Econometrics 2014 Vol. 178 No. 3 P. 444-455

This paper discusses the solution of nonlinear integral equations with noisy integral kernels as they appear in nonparametric instrumental regression. We propose a regularized Newton-type iteration and establish convergence and convergence rate results. A particular emphasis is on instrumental regression models where the usual conditional mean assumption is replaced by a stronger independence assumption. We ...

Added: October 12, 2016

Veretennikov A., Working papers by Cornell University. Series cond-mat.soft "arxiv.org" ( 2014 P. 1-12

THIS ITEM should be deleted: it is a preprint, not an article. ...

Added: December 16, 2014

Veretennikov A., Veretennikova M., / Cornell University. Series "Working papers by Cornell University". 2019.

New convergence rate asymptotic bound for a class of homogeneous Markov chains is established. ...

Added: November 14, 2019