Large deviations of a class of non-homogeneous Markov chains

Let Σ = {lcub}1, 2, …, r{rcub} be a finite set of points. Let P_n = {lcub}p_n( i, j ) : i, j ∈ Σ{rcub} be an r x r stochastic matrix for n ≥ 1, and $p$ be a distribution on Σ. Let now be the (non-homogeneous) Markov measure on the sequence space $S\infty$ with Borel sets corresponding to initial distribution $p$ and transition kernels {lcub}P_n{rcub}.; We now describe the class of non-homogeneous process focused upon in the article. These are the Markov chains where the transition kernels are asymptotically close to a fixed stochastic matrix. Let $p$ be a distribution and P be a stochastic matrix on Σ. Define the collection $A=Ap,P$ by A=&cubl0;Pp &parl0;&cubl0;Pn&cubr0;&parr0; :lim n→∞Pn =P&cubr0;. The collection $A$ can be thought of as perturbations of the stationary Markov chain run with P, and is a natural class in which to explore how “non-homogeneity” enters into the large deviation picture.; Let now f : Σ → $Rd$ be a d ≥ 1 dimensional function. Let also be a “perturbed” non-homogeneous Markov measure. In terms of the coordinate process, define the additive sums Z_n = Z_n(f) for n ≥ 1 by Zn=1n i=1