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We study some limit properties of the harmonic mean of random transition probability for a second-order nonhomogeneous Markov chain and a nonhomogeneous Markov chain indexed by a tree. As corollary, we obtain the property of the harmonic mean of random transition probability for a nonhomogeneous Markov chain.doi:10.1155/2009/503203 fatcat:lt4a4ymdbzdlzltkz67k3yy3lm