Let Σ = {lcub}1, 2, …, r{rcub} be a finite set of points. Let Pn = {lcub}pn( i, j ) : i, j ∈ Σ{rcub} be an r x r stochastic matrix for n ≥ 1, and be a distribution on Σ. Let now be the (non-homogeneous) Markov measure on the sequence space with Borel sets corresponding to initial distribution and transition kernels {lcub}Pn{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 be a distribution and P be a stochastic matrix on Σ. Define the collection by A=&cubl0;Pp&parl0;&cubl0;Pn&cubr0;&parr0; :limn→∞Pn=P&cubr0;.The collection 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 → 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 Zn = Zn(f) for n ≥ 1 byZn=1ni=1>nf&parl0;Xi&parr0;.The goal of this paper is to understand the large deviation behavior of the induced distributions of {lcub}Zn : n ≥ 1{rcub} with respect to . An immediate question which comes to mind asks whether these large deviations differ from the deviations with respect to the stationary chain run with P. The general answer found in our work is “yes” and “no,” and as might be suspected depends on the rate of convergence Pn → P and the structure of the limit matrix P.; More specifically, when P is an irreducible matrix, it turns out that the large deviation of behavior of {lcub}Zn{rcub} under is exactly that under the stationary chain associated with P no matter the r
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机译：令Σ= {lcub} 1,2，…， r {rcub}是一个有限的点集。令 P n = {lcub} p n （ i ， j ）： i ， j ∈Σ{rcub}是 r x r 随机矩阵对于 n ≥1，而展开▼
