## Electronic Journal of Probability

### Approximation of Markov semigroups in total variation distance

#### Abstract

In this paper, we consider Markov chains of the form $X^{n}_{(k+1)/n}=\psi _{k}(X^{n}_{k/n},Z_{k+1}/\sqrt{n},1/n)$ where the innovation comes from the sequence $Z_{k},k\in \mathbb{N} ^{\ast }$ of independent centered random variables with arbitrary law. Then, we study the convergence $\mathbb{E} [f(X^{n}_t)]\rightarrow \mathbb{E} [f(X_t)]$ where $(X_t)_{t \geqslant 0}$ is a Markov process in continuous time. This may be considered as an invariance principle, which generalizes the classical Central Limit Theorem to Markov chains. Alternatively (and this is the main motivation of our paper), $X^{n}$ may be an approximation scheme used in order to compute $\mathbb{E} [f(X_t)]$ by Monte Carlo methods. Estimates of the error are given for smooth test functions $f$ as well as for measurable and bounded $f.$ In order to prove convergence for measurable test functions we assume that $Z_{k}$ satisfies Doeblin’s condition and we use Malliavin calculus type integration by parts formulas based on the smooth part of the law of $Z_{k}$. As an application, we will give estimates of the error in total variation distance for the Ninomiya Victoir scheme.

#### Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 12, 44 pp.

Dates
Accepted: 27 December 2015
First available in Project Euclid: 17 February 2016

https://projecteuclid.org/euclid.ejp/1455717196

Digital Object Identifier
doi:10.1214/16-EJP4079

Mathematical Reviews number (MathSciNet)
MR3485354

Zentralblatt MATH identifier
1338.60097

#### Citation

Bally, Vlad; Rey, Clément. Approximation of Markov semigroups in total variation distance. Electron. J. Probab. 21 (2016), paper no. 12, 44 pp. doi:10.1214/16-EJP4079. https://projecteuclid.org/euclid.ejp/1455717196

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