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.


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