A law of the iterated logarithm for random walk in random scenery with deterministic normalizers

Abstract

LetX, X i ,i≥1, be a sequence of independent and identically distributed ℤ d -valued random vectors. LetS o=0 and $$S_n = \sum\nolimits_{i = 1}^n {X_i } $$ forn≤1. Furthermore letY, Y(α), α∈ℤ d , be independent and identically distributed ℝ-valued random variables, which are independent of theX i . Let $$Z_n = \sum\nolimits_{i = 0}^n {Y(S_i )} $$ . We will call (Z n ) arandom walk in random scenery. In this paper, we consider the law of the iterated logarithm for random walk in random scenery where deterministic normalizers are utilized. For example, we show that if (S n ) is simple, symmetric random walk in the plane,E[Y]=0 andE[Y 2]=1, then $$\mathop {\overline {\lim } }\limits_{n \to \infty } \frac{{Z_n }}{{\sqrt {2n\log (n)\log (\log (n))} }} = \sqrt {\frac{2}{\pi }} a.s.$$