Large deviations for empirical measures of self-interacting Markov chains

Abstract

Let ${\Delta }^o$ be a finite set and, for each probability measure $m$ on ${\Delta }^o$, let $G\left(m\right)$ be a transition kernel on ${\Delta }^o$. Consider the sequence $\left\{X_n\right\}$ of ${\Delta }^o$-valued random variables such that, given $X_0,\dots ,X_n$, the conditional distribution of $X_{n+1}$ is $G\left(L^{n+1}\right)(X_n,\cdot )$, where $L^{n+1}=\frac{1}{n+1}{\sum }_{i=0}^n{\delta }_{X_i}$. Under conditions on $G$ we establish a large deviation principle for the sequence $\left\{L^n\right\}$. As one application of this result we obtain large deviation asymptotics for the Aldous et al. (1988) approximation scheme for quasi-stationary distributions of finite state Markov chains. The conditions on $G$ cover other models as well, including certain models with edge or vertex reinforcement.