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A limit theorem for a class of inhomogeneous Markov processes. (English) Zbl 0687.60070
Let \(\{X(t): t\in\mathbb{R}^+\text{ or }I^+\}\) be a continuous-time or discrete-time irreducible Markov process with a finite state space \(S=\{1,2,\ldots,a\}\). The paper investigates such processes with transition rates of the type
\[ q_{ij}(t)=p(i,j)(\lambda (t))^{U(i,j)}\text{ if } j\ne i\text{ and } q_{ij}(t) =1_{\text{(discrete time)}}- \sum_{k\ne i} q_{ik}(t)\text{ if } j=i, \]
where \(\lambda(t)\) is a suitable rate function with \(\lim_{t\to \infty}\lambda(t)=0\), \(P=(p(i,j))\) is a matrix with \(p(i,j)\ge 0\) for \(i\ne j\) and \(U: S\times S\to [0,\infty)\) is a “cost” function which measures the degree of “reachability” from one state to another. For such processes, using the Kolmogorov forward equations, one proves that there exist positive constants \(\beta_i\), \(i\in S\), such that
\[ \lim_{t\to \infty}P\{X(t)=i\}/(\lambda (t))^{h(i)} = \beta_ i\text{ for each } i\in S, \]
where \(h(i)\) is the so-called “height”; and
\[ P\{X(t)\in \underline s\} = 1+O((\lambda(t))^b),\quad t\to \infty, \]
where \(\underline s =\{i\in S: h(i)=0\}\) and \(b=\min_{i\not\in \underline s}h(i)\).

MSC:
60J27 Continuous-time Markov processes on discrete state spaces
60F05 Central limit and other weak theorems
60F10 Large deviations
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