Oberseminar Stochastik und Data Science : Pavlos Zoubouloglou (UNC Chapel Hill)
Title: 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 $G(m)$ be a transition kernel on $\Delta^o$. Consider the sequence $\{X_n\}$ of $\Delta^o$-valued random variables such that, and given $X_0,\ldots,X_n$, the conditional distribution of $X_{n+1}$ is $G(L^{n+1})(X_n,\cdot)$, where $L^{n+1}=\frac{1}{n+1}\sum_{i=0}^{n}\delta_{X_i}$. Under conditions on $G$ we establish a large deviation principle for the sequence $\{L^n\}$. As one application of this result we obtain large deviation asymptotics for the Aldous-Flannery-Palacios (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.