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)
60G50 Sums of independent random variables; random walks
Full Text: DOI