Results on limiting sets of Markov set chains
Let \(K\) denote a non-empty compact set of \((n \times n)\) stochastic matrices. Let \(S_ 0\) be a compact set of \(1 \times n\) stochastic vectors and define recursively for \(k \geq 1\), \(S_ k=S_{k-1} A=\{xA:x \in S_{k-1}\) and \(A \in K\}\). \(\{S_ k\}\), \(k \geq 0\), is called a Markov set chain. The coefficient of ergodicity \(\tau_ 1(A)\) for any stochastic \(A\) is defined by \(\tau_ 1(A)=\sup \{ \| \delta A \|_ 1\), \(\| \delta \|_ 1=1\), \(\delta{\underset\sim 1}=0\}\). This satisfies \(\tau_ 1(A) \leq 1\) and is simply expressed in terms of the entries of \(A\) [reviewer, Nonnegative matrices and Markov chains (1981; Zbl 0471.60001)]. Put \(\overline \tau_ 1=\max_{A \in K} \tau_ 1(A)\). Define diameter for a compact set \(G\) of stochastic matrices by \(d(G)=\max_{A,B \in G} \| A-B \|_ 1\). It is known [reviewer, Stat. Probab. Lett. 2, 159-163 (1984; Zbl 0541.15009)] that if \(\overline \tau_ 1<1\), then \(\lim_{k \to \infty}S_ k\) exists, in the sense of convergence in the Hausdorff metric \(\Delta\) on compact sets; denote the limiting set by \(S_ \infty\). In this paper the author shows further that if \(\overline \tau_ 1<1\), then \(d(S_ \infty) \leq (1-\overline \tau_ 1)^{-1}d(K)\). Further, if \(K'\) is any other compact set of stochastic matrices, and \(\{S_ k'\}\), \(k \geq 0\), is the corresponding Markov set chain, with \(\overline \tau_ 1'<1\), then \(\Delta(S_ \infty,S_ \infty') \leq (1-\overline \tau_ 1)^{-1} \Delta (K,K')\). In the sense that \((1-\overline \tau_ 1)^{-1}\) gives a bound on the magnification error \(\Delta(S_ \infty, S_ \infty')/ \Delta(K,K')\), it is a condition number. This last result in a simpler setting occurs in articles of the reviewer [for example: Adv. Appl. Probab. 20, No. 1, 228-230 (1988; Zbl 0645.60070)].
scientific article; zbMATH DE number 519694