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## 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.

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 be applicable when one transition can not guarantee any convergence. Moreover, a better estimate can be obtained for a higher number of transitions steps. A law of large numbers is presented for a class of ergodic nonlinear Markov chains with finite state space that may serve as a basis for nonparametric estimation and other statistics.

Shchegolev A., A new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chains / 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

Nikitin Y. Y., Petrov V. V., Zaitsev A. Y. et al., Vestnik of the St. Petersburg University: Mathematics 2018 Vol. 51 No. 2 P. 201-232

This is the first in a series of reviews devoted to the scientific achievements of the Leningrad–St. Petersburg school of probability and statistics in the period from 1947 to 2017. It is devoted to limit theorems for sums of independent random variables—a traditional subject for St. Petersburg. It refers to the classical limit theorems: the ...

Added: October 1, 2019

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

Tkachenko A., Esaulov D., Applied Mathematical Modelling 2016

This paper is focused on a multichannel queueing system with heterogeneous servers, regenerative input flow, and balking. Service times are random variables but not necessary exponential. If a new customer encounters j other customers in the system, then it joins the queue with probability fj and leaves with probability 1-fj . For this system the ...

Added: October 19, 2016

Kroshnin A., Journal of Convex Analysis 2018 Vol. 25 No. 4 P. 1371-1395

We consider the space P(X) of probability measures on arbitrary Radon space X endowed with a transportation cost J(μ, ν) generated by a nonnegative continuous cost function. For a probability distribution on P(X) we formulate a notion of average with respect to this transportation cost, called here the Fréchet barycenter, prove a version of the law ...

Added: November 23, 2018

Tkachenko A., Moscow University Mathematics Bulletin 2013 Vol. 68 No. 2 P. 93-97

This paper is devoted to $M|GI|1|\infty$ queueing system with unreliable server and customer service times depending on the system state. Condition of ergodicity and generating function are found in the stationary state. ...

Added: March 13, 2014

Blank M., Nonlinearity 2014 Vol. 27 No. 5 P. 953-971

We discuss conditions for unique ergodicity of a collective random walk on a continuous circle. Individual particles in this collective motion perform independent (and different in general) random walks conditioned by the assumption that the particles cannot overrun each other. Additionally to sufficient conditions for the unique ergodicity we discover a new and unexpected way ...

Added: November 21, 2014

А. В. Ткаченко, Вестник Московского университета. Серия 1: Математика. Механика 2013 № 2 С. 12-17

This paper is devoted to $M|GI|1|\infty$ queueing system with unreliable server and customer service times depending on the system state. Condition of ergodicity and generating function are found in the stationary state. ...

Added: March 27, 2013

Goetze F., Naumov A.A., Tikhomirov A., Bernoulli: a journal of mathematical statistics and probability 2018 Vol. 24 No. 3 P. 2358-2400

We consider a random symmetric matrix X=[X_{jk}]_{j,k=1}^n with upper triangular entries being i.i.d. random variables with mean zero and unit variance. We additionally suppose that \E|X_{11}|^{4+\delta}=:\mu_{4+\delta}<\infty for some \deta>0. The aim of this paper is to significantly extend a recent result of the authors Götze, Naumov and Tikhomirov (2015) and show that with high probability the typical ...

Added: February 13, 2018

Afanasyeva L. G., Tkachenko A., Theory of Probability and Its Applications 2014 Vol. 58 No. 2 P. 174-192

We consider the multichannel queueing system with nonidentical servers and regenerative input flow. The necessary and sufficient condition for ergodicity is established, and functional limit theorems for high and ultra-high load are proved. As a corollary, the ergodicity condition for queues with unreliable servers is obtained. Suggested approaches are used to prove the ergodic theorem ...

Added: August 20, 2014

Ulyanov V. V., Bobkov S., Danshina M., On rate of convergence to the Poisson law of the number of cycles in the generalized random graphs / . 2021. No. 21027.

Convergence of order O(1/ √ n) is obtained for the distance in total variation between the Poisson distribution and the distribution of the number of fixed size cycles in generalized random graphs with random vertex weights. The weights are assumed to be independent identically distributed random variables which have a power-law distribution. The proof is ...

Added: March 29, 2021

Dragunova K., Гаращенкова А. А., Remizov I., Numerical Study of the Rate of Convergence of Chernoff Approximations to Solutions of the Heat Equation / Cornell University. Series arXiv "math". 2021.

Chernoff approximations are a flexible and powerful tool of functional analysis, which can be used, in particular, to find numerically approximate solutions of some differential equations with variable coefficients. For many classes of equations such approximations have already been constructed, however, the speed of their convergence to the exact solution has not been properly studied. ...

Added: December 16, 2021

Tkachenko A., Multichannel queuing systems with balking and regenerative input flow / Высшая школа экономики. Series WP BRP "Science, Technology and Innovation". 2013. No. 14.

Motivated by the application to telephone call centers this paper is focused on the multichannel queueing system with heterogeneous servers, regenerative input flow and balking. Servers times are random variables but not necessary exponential. If a new customer encountering j other customers in the system then it stays for service with probability fj and gets ...

Added: August 1, 2013

Veretennikov A., Veretennikova M., Известия РАН. Серия математическая 2022 Т. 86 № 1 С. 98-133

We continue the work of improving the rate of convergence of ergodic homogeneous Markov chains. The setting is more general than in previous papers: we are able to get rid of the assumption about a common dominating measure and consider the case of inhomogeneous Markov chains as well as more general state spaces. We give examples ...

Added: March 14, 2022

Ткаченко А.В., Вестник Московского университета. Серия 1: Математика. Механика 2014 № 1 С. 53-57

This paper is focused on multichannel queueing system with heterogeneous servers and regenerative input flow in a random environment. The environment can destroy all the system and then system is reconstructed. Ergodicity condition of the system is obtained. ...

Added: May 11, 2013

Povolotsky A. M., Journal of Statistical Mechanics: Theory and Experiment 2019 No. 074003 P. 1-22

We establish the exact laws of large numbers for two time additive quantities in the raise and peel model, the number of tiles removed by avalanches and the number of global avalanches happened by given time. The validity of conjectures for the related stationary state correlation functions then follow. The proof is based on the ...

Added: October 8, 2019

Veretennikov A., Veretennikova M., On convergence rate for homogeneous Markov chains / 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

Decrouez G. G., Borovkov K., Gilson M., Journal of Applied Probability 2014 Vol. 51 No. 3 P. 837-857

The paper deals with nonlinear Poisson neuron network models with bounded memory dynamics, which can include both Hebbian learning mechanisms and refractory periods. The state of the network is described by the times elapsed since its neurons fired within the post-synaptic transfer kernel memory span, and the current strengths of synaptic connections, the state spaces ...

Added: September 29, 2014

Kroshnin A., Sobolevski A., Fréchet Barycenters and a Law of Large Numbers for Measures on the Real Line / Cornell University. Series arXiv "math". 2015. No. 1512.08421.

Endow the space P(R) of probability measures on R with a transportation cost J(mu, nu) generated by a translation-invariant convex cost function. For a probability distribution on P(R) we formulate a notion of average with respect to this transportation cost, called here the Fréchet barycenter, prove a version of the law of large numbers for ...

Added: December 31, 2015

Tkachenko A., Moscow University Mathematics Bulletin 2014 Vol. 69 No. 1 P. 37-40

This paper is focused on a multichannel queueing system with heterogeneous servers and regenerative input flow operating in a random environment. The environment can destroy the whole system and the system is reconstructed after that. The necessary and sufficient ergodicity condition is obtained for the system. ...

Added: August 20, 2014

Blank M., Доклады Академии наук 2013 Т. 448 № 6 С. 629-632

We give conditions for unique ergodicity for a discrete time collective
random walk on a continuous circle. Individual particles in this collective
motion perform independent (and different) random walks conditioned
by the assumption that the particles cannot overrun each other.
Deterministic version of this system is studied as well. ...

Added: November 25, 2014

Kleptsyn V., Alvarez S., Malicet D. et al., Groups with infinitely many ends acting analytically on the circle / Cornell University. Series math "arxiv.org". 2015.

Added: June 22, 2016

Litvin Y. V., Абрамов И. В., Технологии техносферной безопасности 2016 № 66

Advanced approach to the assessment of a random time of arrival fire fighting calculation on the object of protection, the time of their employment and the free combustion. There is some quantitative assessments with the review of analytical methods and simulation ...

Added: August 27, 2016

Furmanov K. K., Nikol'skii I. M., Computational Mathematics and Modeling 2016 Vol. 27 No. 2 P. 247-253

Added: December 22, 2016