## A zero or one law for one dimensional random walks in random environments.(English)Zbl 0642.60022

For each $$z\in {\mathbb{Z}}$$ let $$\Gamma_ x$$ denote the set of all probability measures on Z, and put $$\Gamma =\prod_{x\in {\mathbb{Z}}}\Gamma_ x$$. $$\Gamma$$ is endowed with the product topology, derived from the topology of weak convergence on the coordinate sets, and with the associated Borel $$\sigma$$-algebra. In terms of a probability measure M on $$\Gamma$$ and a family $$(p_ s(x,y))_{s\in \Gamma}$$, x,y$$\in {\mathbb{Z}}$$, of transition probabilities, a random walk in random environment starting at 0, $$(X_ n)_{n\in {\mathbb{Z}}_+}$$, is defind on $${\mathbb{Z}}^{{\mathbb{Z}}_+}$$. Under mild regularity conditions on M and $$(p_ s)_{s\in \Gamma}$$, the author proves mainly the following results.
Theorem 1. If A is a shift invariant and translation invariant event, then $$P(A)=0$$ or 1.
Theorem 2. Let $$B\subset {\mathbb{Z}}$$. Then $$P(X_ n\in B$$ $$i.o.)=1$$ in three cases: a) P(lim $$X_ n=\infty)=1$$ and $$B\cap {\mathbb{Z}}_+$$ is infinite; b) P(lim $$X_ n=-\infty)=1$$ and $$B\cap {\mathbb{Z}}_-$$ is infinite; c) P(lim sup $$X_ n=\infty$$, lim inf $$X_ n=-\infty)=1$$ and $$B\neq \emptyset$$. In all other cases $$P(X_ n\in B$$ $$i.o.)=0$$.
Reviewer: A.Spătaru

### MSC:

 60F20 Zero-one laws 60G50 Sums of independent random variables; random walks 60K99 Special processes 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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