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The Poisson boundary for homogeneous random walks. (English. Russian original) Zbl 0939.60030
Russ. Math. Surv. 54, No. 2, 441-442 (1999); translation from Usp. Mat. Nauk 54, No. 2, 177-178 (1999).
Let $$\mathcal L$$ be a homogeneous Markov chain with a countable space $$X$$. $$X^\infty =\{(\alpha _n)_{n=0}^\infty , \alpha _n\in X \}$$ is the space of paths of $$\mathcal L$$, and $$\mathbf P$$ is a probability masure on $$X^\infty$$. If $$\mathcal L$$ is transient, irreducible and aperiodic, there exists an essentially unique countable decomposition $$\Delta =\{C_0, C_1, \dots\}$$ of disjoined sets of positive probability which are invariant w.r.t. shifts $$T^{-1}C=C$$, $$T(\alpha _0, \alpha _1,\dots)=(\alpha _1, \alpha _2, \dots)$$. Poisson boundary of $$\mathcal L$$ is a factor space $$(X^\infty , {\mathbf P})/\Delta$$. The Poisson boundary is trivial if $$\Delta$$ consists of a single set that is an atom. Then, the Poisson boundary for transient, irreducible and aperiodic random walks in $$\mathbb Z^2$$ and $$\mathbb Z_+\times \mathbb Z$$ is trivial. The structure of Poisson boundary for random walks in $$(\mathbb Z_+)^2$$ is given in terms of the mean drift vectors. Some generalizations are presented for random walks in $$(\mathbb Z_+)^n$$ and $$(\mathbb Z_+)^n\times \mathbb Z^m$$.

##### MSC:
 60G50 Sums of independent random variables; random walks 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
##### Keywords:
homogeneous random walk; Poisson boundary
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