# zbMATH — the first resource for mathematics

On a solution of Kesten’s problem. (English. Russian original) Zbl 0722.60072
Theory Probab. Appl. 35, No. 2, 360-363 (1990); translation from Teor. Veroyatn. Primen. 35, No. 2, 358-361 (1990).
Using the results of himself [ibid. 33, No.2, 228-238 (1988); resp. ibid. 33, No.2, 246-256 (1988; Zbl 0656.60077)] and taking an idea of F. Solomon [Ann. of Probab. 3, 1-31 (1975; Zbl 0305.60029)], the author gives a solution of Kesten’s problem with $$R=1$$ and $$L=2:$$ A necessary and sufficient condition that there is such a $$c>0$$ that $$\lim_{t\to \infty}x(t)/t=c$$ is $$E_ pr_ -<+\infty$$, where $$\{$$ x(t), $$t=0,1,...\}$$ is a random walk on the one-dimensional lattice $$Z^ 1$$ in Bernoulli random medium, which at a particle $$x\in Z^ 1$$ may walk to any one of x-2, x-1, and $$x+1$$ after each unit-time, while $$p\triangleq \{p(x)$$, $$x\in Z^ 1\}$$ is i.i.d. with the same continuous distribution at $$[\epsilon_ 0,1-\epsilon_ 0]$$, $$\epsilon_ 0>0$$ is sufficiently small, and $$r_ -$$ a sum of some random series.
Reviewer: Y.Ge (Beijing)
##### MSC:
 60G50 Sums of independent random variables; random walks
##### Keywords:
Kesten’s problem; Bernoulli random medium
Full Text: