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


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