## The local time of a random walk in a random environment.(English)Zbl 0623.60092

New perspectives in theoretical and applied statistics, Sel. Pap. 3rd Int. Meet. Stat., Bilbao/Spain 1986, 503-518 (1987).
[For the entire collection see Zbl 0608.00013.]
A sequence $$X=\{X_ i$$; $$i\in {\mathbb{Z}}\}$$ of i.i.d. random variables $$X_ i$$ taking values in ]0,1[ is called a random environment. The random walk is defined by $$S_ 0=0$$ and $$P_ X(S_{n+1}=i+1| S_ n=i)=X_ i$$, $$P_ X(S_{n+1}=i-1| S_ n=i)=1-X_ i$$ $$(n=0,1,2,...)$$. In a recent paper P. Deheuvels and the author [Probab. Theory Relat. Fields 72, 215-230 (1986; Zbl 0572.60070)] obtained strong limiting bounds for th maximum reached by the random walk and for its local time given by $$\xi (x,n)=\#\{k:$$ $$0\leq k\leq n$$, $$S_ k=x\}.$$
In this paper the author obtains further properties of the maximum, in particular estimations for its distribution function, and shows that the bounds for $$\xi$$ obtained in the earlier paper are not far from the best possible ones.
Reviewer: M.Dozzi

### MSC:

 60G50 Sums of independent random variables; random walks 60J55 Local time and additive functionals

### Keywords:

Polya’s theorem; random environment; local time

### Citations:

Zbl 0608.00013; Zbl 0572.60070