Some limit theorems for ratios of transition probabilities (Q1090004)

In this paper, the author considers infinite-dimensional forward convergent stochastic chains. Among other results, she proves an infinite-dimensional version of the following theorem of the reviewer, \textit{A. Nakassis} and \textit{D. Issacson} [Stat. Decis. 2, 363-375 (1984; Zbl 0549.60060)]: For a finite-dimensional convergent chain \((P_ n)\) with basis \(\{T,C_ 1,C_ 2,...,C_ p\}\), we have for any \(L\subset T\) and for each positive integer k, \[ \lim_{m\to \infty}(\sum^{m}_{n=k}(P_ n)_ Tc_ L)/(\sum^{m}_{n=k}(P_ n)_{LL^ c})=0, \] where \((P_ n)_{AB}=\sum_{i\in A,j\in B}(P_ n)_{ij}\), and the term ''basis'' has the same meaning as in the quoted paper above.