Liu, Wen A strong limit theorem for the harmonic mean of the random transition probabilities of finite nonhomogeneous Markov chains. (Chinese. English summary) Zbl 0964.60024 Acta Math. Sci. (Chin. Ed.) 20, No. 1, 81-84 (2000). Summary: Let \(\{X_n, n\geq 0\}\) be a nonhomogeneous Markov chain with state space \(I= \{1,\dots, N\}\) and transition probabilities \(p_n(i,j)= P(X_n= j\mid X_{n-1}= i)\). Conditions that the harmonic mean of random transition probabilities \(p_1(X_0, X_1),\dots, p_n(X_{n-1}, X_n)\) converges to \(1/N\) a.s. are investigated, and the differentiation technique on a net together with the tool of generating function for the study of strong limit theorems of Markov chains is presented. Cited in 1 Document MSC: 60F15 Strong limit theorems 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) Keywords:random transition probability; harmonic mean; differentiation on a net; strong limit theorem PDFBibTeX XMLCite \textit{W. Liu}, Acta Math. Sci. (Chin. Ed.) 20, No. 1, 81--84 (2000; Zbl 0964.60024)