Symmetry characterizations of certain distributions. 3. Discounted additive functional and large deviations for a general finite-state Markov chain

Abstract

In Baxter and Williams [1] we began a study of Abel averages,as opposed to the oft-studied Cesàro averages In Baxter and Williams [2], hereinafter referred to as [BW2], we studied the large-deviation behaviour of these averages. In the case where X is an irreducible Markov chain on a finite state-space S = {1,…, n}, we observed thatandwhere π is the invariant distribution of X. We noted thatwhere v is an n-vector, δ(v):= sup{Re(z): z ∈ spect(Q+ V)}, and where spect(·) denotes spectrum (here the set of eigenvalues), Q is the Q-matrix of X, and V denotes the diagonal matrix diag (vi). It is also true that the large-deviation property holds for Ct with rate function I denned on M = {(xt)i ∈ S: xi ≥ 0, Σixi = 1}.