Empirical measure large deviations for reinforced chains on finite spaces

Let A be a transition probability kernel on finite state space Δo={1,…,d} such that A(x,y)>0 for all x,y∈Δo. Consider reinforced chain given as sequence {Xn,n∈N0} of Δo-valued random variables, defined recursively according to, Ln=1n∑i=0n−1δXi,P(Xn∈⋅∣X0,…,Xn−1)=LnA(⋅). We establish large deviation principle {Ln,n∈N}. The rate function takes strikingly different form than the Donsker–Varadhan associated with empirical measure Markov and is described in terms novel deterministic infinite horizon discounted cost control problem an linear controlled dynamics nonlinear running involving relative entropy function. Proofs are based analysis time-reversal representations log-transforms exponential moments, weak convergence methods.