The paper under review extends the notions of stochastic matrix, stable matrix and definite set. The author defines the generalized stochastic matrix as a nonnegative scalar product of a usual stochastic matrix; a nonnegative matrix \(P\) is \([\Delta]\)-stable if \(P^L_K\) is a generalized stochastic matrix for any partition in \(\Delta\) and any \(L\) in \(\Sigma\); a nonnegative matrix is \(\Delta\)-stable if \(\Delta\) is the least fine partition for which \(P\) is a \([\Delta]\)-stable. Consider a finite Markov chain with state space \(S=\{1,2,\dots,r\}\), an initial distribution \(p_0\) and transition matrices \((P_n)_{n\geq 1}\). Define \(P_{m,n}=P_{m+1} P_{m+2}\cdots P_n\). Then the chain \(P_{m, n}\) is weakly \([\Delta]\)-ergodic if and only if there exist \([\Delta]\)-stable \(r\times p\) matrices \(\Pi_{m,n} (m<n)\) such that \[ \lim_{n\to \infty} (P_{m, n}^+ - \Pi_{m, n}) = 0. \] The \([\Delta]\)-stable matrices on \(\Sigma\) play a basic role in the general \(\Delta\)-ergodic theory. A few examples are given in Section 1 to illustrate the generalized notions of \([\Delta]\)-stable and \(\Delta\)-stable. A nonempty collection \(D=\{P_1,P_2,\dots,P_t\}\) is a \(k\)-definite set if \(P_{i1}P_{i2}\cdots P_{il}\) is a stable matrix for any \(l\geq k\) and \(k\) is the smallest number with this property.
In Section 2, the author defines \(G_{\Delta_1,\Delta_2}\) as a collection of stochastic matrices which are \([\Delta_1]\)-stable matrices on \(\Delta_2\). A linked structure of the product of \(P_1P_2\cdots P_n\) is given. Theorem 2.10 allows to determine if a finite set of matrices is \(k\)-definite or a Markov chain has a finite convergence time. Remark 2.13 gives the characterisation of the linked structure. Many examples from concrete matrices are given to illustrate the different stability notions and a few results are quoted from the author’s previous work, as well as a challenging problem at the end of the article. It would have been nice to see practical examples and real motivation for the developed concepts in this paper, specifically, the use of the ergodicity coefficients.