Searcharxiv⌕ Search

arXiv subjects

Alexander Veretennikov

Publications and source records attributed to Alexander Veretennikov.

At least 19 recordsLinked to original sources
arXiv2026

On efficient estimates of the rate of convergence for Markov chains

The paper presents efficient approaches for evaluating convergence rate in total variation for finite and general linear Markov chains. The motivation for studying convergence rate in this metric is its usefulness in various limit theorems. For homogeneous Markov chains the goal is to compare several different methods: (1) the second eigenvalue for the transition matrix method (the method no. 1), (2) the method based on Markov -- Dobrushin's ergodic coefficient, and the new spectral method developed in earlier works, as well as modifications of they both by iterations (the ``other methods''). We answer the question whether or not the ``other methods'' may provide the optimal or close to optimal convergence rate in the case of homogeneous Markov chains. The answer turns out to be positive for appropriate modifications of both ``other methods''. The analogues of these ``other methods'' for the non-homogeneous Markov chains are also presented. The work is theoretical. However, the methods of computing efficient bounds of convergence rates may be in demand in various applied areas.

math.PR↗
arXiv2025

On nonlinear weak law of large numbers

A new version of a weak nonlinear law of large numbers proposed. The existence of the first moment for any summand is not assumed. The assumption of independence is understood in the nonlinear sense, and may be further a little relaxed.

math.PR↗
arXiv2023

On higher order moments and recurrence of an SDE with switching

Second order recurrence of a $d$-dimensional diffusion with an additive Wiener process, with switching, and with one recurrent and one transient regime and constant switching intensities is established under suitable conditions. The approach is based on embedded Markov chains and a priori bounds for the moments of $X_t$ at moments of jumps of the discrete component, as well as on some simple martingale properties.

math.PR↗
arXiv2022

On convergence rate bounds for a class of nonlinear Markov chains

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. The new convergence rate may be used, in particular, for the justification of $D$-condition in the Extreme Values theory.

math.PR↗
arXiv2020

Note on local mixing techniques for stochastic differential equations

This paper discusses several techniques which may be used for applying the coupling method to solutions of stochastic differential equations (SDEs). They all work in dimension $d\ge 1$, although, in $d=1$ the most natural way is to use intersections of trajectories, which requires nothing but strong Markov property and non-degeneracy of the diffusion coefficient. In dimensions $d>1$ it is possible to use embedded Markov chains either by considering discrete times $n=0,1,\ldots$, or by arranging special stopping time sequences and to use local Markov -- Dobrushin's (MD) condition. Further applications may be based on one or another version of the MD condition. For studies of convergence and mixing rates the (Markov) process must be strong Markov and recurrent; however, recurrence is a separate issue which is not discussed in this paper.

math.PR↗
arXiv2020

Yet again on iteration improvement for averaged expected cost control for 1D ergodic diffusions

The paper is a full version of the short presentation in \cite{amv17}. Ergodic control for one-dimensional controlled diffusion is tackled; both drift and diffusion coefficients may depend on a strategy which is assumed markovian. Ergodic HJB equation is established and existence and uniqueness of its solution is proved, as well as the convergence of the reward improvement algorithm.

math.PR↗
arXiv2020

On pathwise uniqueness for multidimensional McKean--Vlasov equations

Pathwise uniqueness for multi-dimensional stochastic McKean--Vlasov equation is established under moderate regularity conditions on the drift and diffusion coefficients. Both drift and diffusion depend on the marginal measure of the solution. For pathwise uniqueness, the drift is assumed to be Dini-continuous in the state variable, while the diffusion must be Lipschitz, continuous in time and uniformly nondegenerate. The setting is classical McKean--Vlasov, that is, coefficients of the equation are represented as integrals over the marginal distributions of the process.

math.PR↗
arXiv2020

On polynomial recurrence for reliability system with a warm reserve

Conditions for positive and polynomial recurrence have been proposed for a class of reliability models of two elements with transitions from working state to failure and back. As a consequence, uniqueness of stationary distribution of the model is proved; the rate of convergence towards this distribution may be theoretically evaluated on the basis of the established recurrence.

math.PR↗
arXiv2019

On convergence rate for homogeneous Markov chains

Improved rates of convergence for ergodic homogeneous Markov chains are studied. In comparison to the earlier papers the setting is also generalised to the case without a unique dominated measure. Examples are provided where the new bound is compared with the classical Markov -- Dobrushin inequality and with the second eigenvalue of the transition matrix for finite state spaces.

math.PR↗