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.

https://www.math.fau.de/stochastik/oberseminar-stochastik/