Limit theorems for geometrically ergodic Markov chains

Loïc Hervé (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

Let $(E,\cE)$ be a countably generated state space, let $(X_n)_n$ be an aperiodic and $\psi$-irreducible $V$-geometrically ergodic Markov chain on $E$, with $V : E\r [1,+\infty[$ and $\psi$ a $\sigma$-finite positive measure on $E$. Let $\pi$ be the $P$-invariant distribution, and let $\xi : E\r\R$ measurable and dominated by $\sqrt V$. Then $\sigma^2 = \lim_n n^{-1}\E_x[(S_n)^2]$ exists for any $x\in E$ (and does not depend on $x$), and if $\sigma^2 >0$, then $n^{-\frac{1}{2}}[\xi(X_1)+\ldots+\xi(X_n)-n\pi(\xi)]$ converges in distribution to the normal distribution ${\cal N}(0,\sigma^2)$. In this work we prove that, for any initial distribution $\mu_0$ satisfying $\mu_{_0}(V) < +\infty$ and under the condition $\sigma^2>0$, - If $\xi$ is dominated by $V^\alpha$ with $\alpha<\frac{1}{4}$, then the rate of convergence in the c.l.t is $O(\frac{1}{\sqrt n})$. - If $\xi$ is dominated by $V^\alpha$ with $\alpha<\frac{1}{2}$, then $(\xi(X_n))_n$ satisfies a local limit theorem under a usual non-arithmeticity assumption.

Mots clés: Markov chains : geometric ergodicity : central limit theorems : spectral method

Domaines: Probabilités [math.PR]

Loïc Hervé. Limit theorems for geometrically ergodic Markov chains. 2006. ⟨hal-00008462v2⟩