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


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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