Let Δ be a finite set and, for each probability measure m on Δ, let G(m) be a transition kernel on Δ. Consider the sequence {X_n} of Δ-valued random variables such that, given X_0,...,X_n, the conditional distribution of X_{n+1} is G(L^{n+1})(X_n, . ), where L^{n+1} is the empirical measure at instant n. 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.

Arxiv link: https://arxiv.org/abs/2304.01384