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Recurrence and transience criteria for random walk in a random environment. (English) Zbl 0545.60066

In this paper V. I. Oseledec’s multiplicative ergodic theorem [A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Tr. Mosk. mat. O.-va. 19, 179-210 (1968); English translation in Trans. Mosc. Math. Soc. 19(1968), 197-231 (1969)] is used to give recurrence and transience criteria for random walk in a random environment on the integers. These criteria are obtained by using the Markov property to derive a system of difference equations with random coefficients which then can be analyzed in terms of products of random matrices. The criteria are phrased in terms of the set of Lyapunov exponents of the sequence of random matrices. Some examples are computed. Finally, the entire methodology is used to give analogous criteria for Markov chains on the integers with periodic transition functions.

MSC:

60G50 Sums of independent random variables; random walks
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)

Citations:

Zbl 0236.93034
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