Quelques propriétés de certaines marches aléatoires en milieu aléatoire. (Some properties of certain random walks in random environment). (French) Zbl 0626.60110

Biaz, Khaleb

Publ. Inst. Rech. Math. Rennes 1986, No. 1, 1-40 (1986).

Consider an irreducible random walk on the integer lattice in a random environment. Assuming that the increments are bounded by a natural integer \(A\geq 1\), it is shown that such a random walk is either recurrent or transient. Moreover, recurrence is null-recurrence.
Next, for a transient random walk with \(A=1\) the moments of any order of the first passage time from 0 to 1 given the environment are calculated. Finally, giving up the restriction \(A<\infty\), a central limit theorem is proved assuming a centered environment.
The results obtained concerning recurrence and the central limit theorem slightly extend results of C. C. Heyde, M. Westcott and E. R. Willians [J. Stat. Phys. 28, 375-380 (1982; Zbl 0512.60063)] and F. Solomon [Ann. Probab. 3, 1-31 (1975; Zbl 0305.60029)].

Reviewer: M.Iosifescu

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60F05	Central limit and other weak theorems
60G50	Sums of independent random variables; random walks

Keywords:

random environment; recurrent; transient; first passage time; central limit theorem