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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