## In random environment the local time can be very big.(English)Zbl 0716.60087

Les processus stochastiques, Coll. Paul Lévy, Palaiseau/Fr. 1987, Astérisque 157-158, 321-339 (1988).
[For the entire collection see Zbl 0649.00017.]
Let $${\mathcal E}=\{...,E_{-2},E_{-1},E_ 0,E_ 1,E_ 2,...\}$$ be a sequence of i.i.d. random variables with $$P\{E_ 0<x\}=F(x)$$, $$0<x<1$$, $$F(0)=0$$, $$F(1)=1$$. Such a sequence $${\mathcal E}$$ is called a random environment; any realization of it will be denoted by the same letter $${\mathcal E}$$. For any fixed sample sequence of this environment, define a random walk $$\{R_ n\}$$ by $$R_ 0=0$$ and $P_{{\mathcal E}}\{R_{n+1}=i+1| \quad R_ n=i\}=1-P_{{\mathcal E}}\{R_{n+1}=i- 1| \quad R_ n=i\}=E_ i\quad (n=0,1,...,\quad i=0,\pm 1,...).$ Assume (i) $$P\{a<E_ 0<1-a\}=1$$ for some $$0<a<$$, (ii) E log((1-E$${}_ 0)/E_ 0)=0$$ and (iii) $$0<\sigma^ 2=E(\log ((1-E_ 0)/E_ 0))^ 2<\infty$$. Let $$\xi (x,n)=\#\{k:\;0\leq k\leq n,\quad R_ k=x\},\xi (n)=\max_{x}\xi (x,n)$$. The first result of this paper gives an upper bound for $$\xi$$ (0,n) as follows: $$\xi$$ (0,n)$$\leq \exp ((1-\theta_ n)\log n)$$ a.s. for all but finitely many n, where $$\theta_ n=\exp (- C(\log_ 2 n)(\log_ 3 n)^{-1/2} \log_ 4 n),$$ where $$\log_ p$$ is the pth iterate of log and the meaning of a.s. is: for almost all realizations of $${\mathcal E}$$ the stated inequality holds with $$P_{{\mathcal E}}$$-probability 1. As to the behaviour of $$\xi$$ (n), it is conjectured that $$\limsup_{n\to \infty}n^{-1}\xi (n)=C$$ a.s. for some $$0<C=C(F)<1$$, and in the case of $$P\{E_ i=p\}=P\{E_ i=1-p\}=,$$ $$0<p<$$, it is proved that $$\limsup_{n\to \infty}n^{-1}\xi (n)\geq g(p)$$ a.s., where $$1/g(p)=16f(x)/p+1,$$ $$f(x)=(2x^ 2-x+1)/(1-x)^ 3$$ and $$x=p/(1-p)$$. These results are related to some of P. Deheuvels and the author [Probab. Theory Relat. Fields 72, No.2, 215-230 (1986; Zbl 0572.60070)] and the author [New perspectives in theoretical and applied statistics, Sel. Pap. 3rd Int. Meet. Stat., Bilbao/Spain 1986, 503-518 (1987; Zbl 0623.60092)], where a.s. inequalities are given for describing how small and how large $$\xi$$ (0,n) can be. The paper also contains a number of interesting lemmas on various other aspects of random walks in random environments, as well as in the classical setting.

### MSC:

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

### Citations:

Zbl 0649.00017; Zbl 0572.60070; Zbl 0623.60092