Let $\Delta^o$ be a finite set and, for each probability measure $m$ on $\Delta^o$, let $G(m)$ be a transition probability kernel on $\Delta^o$. Fix $x_0 \in \Delta^o$ and consider the chain $\{ X_n, \; n \in \mathbb{ N}_0\}$ of $\Delta^o$-valued random variables such that $X_0=x$, and given $X_0, \ldots , X_n$, the conditional distribution of $X_{ n+1}$ is $G(L^{ n+1})(X_n, \cdot)$, where $L^{ n+1} = \frac{ 1}{ n+1} \sum_{ i=0}^{ n} \delta_{ X_i}$ is the empirical measure at instant $n$. Under conditions on $G$ we establish a large deviation principle for the empirical measure sequence $\{ L^n, \; n \in \mathbb{ N}\}$. As one application of this result we obtain large deviation asymptotics for the Aldous-Flannery-Palacios (1988) approximation scheme for quasistationary distributions of irreducible finite state Markov chains. The conditions on $G$ cover various other models of reinforced stochastic evolutions as well, including certain vertex reinforced and edge reinforced random walks and a variant of the PageRank algorithm. The particular case where $G(m)$ does not depend on $m$ corresponds to the classical results of Donsker and Varadhan (1975) on large deviations of empirical measures of Markov processes. However, unlike this classical setting, for the general self-interacting models considered here, the rate function takes a very different form; it is typically non-convex and is given through a dynamical variational formula with an infinite horizon discounted objective function.
Categories: math.PR math.OC