1. INTRODUCTION 3

This result is an improvement of that in [5]; there it was required that the Xi

be bounded random variables and the constant was not identified. Theorem 1.1 is

a more precise estimate than is needed for Theorem 1.6; this is why Theorem 1.1

needs to be stated in terms of H(n) while Theorem 1.6 does not.

For an LIL for −Rn we have a different rate.

Theorem 1.7. We have

lim sup

n→∞

−Rn

n log log n/(log n)2

=

(2π)−2

√

det Γ κ(2,

2)−4,

a.s.

The study of the range of a lattice-valued (or Zd-valued) random walk has a long

history in probability and the results show a strong dependence on the dimension

d. See Section 2 for a brief history of the literature. The two dimensional case

seems to be the most diﬃcult; in one dimension no renormalization is needed (see

[9]), while for d ≥ 3 the tails are sub-Gaussian and have asymptotically symmetric

behavior. In two dimensions, renormalization is needed and the tails have non-

symmetric behavior. In this case, the central limit theorem was proved in 1986 in

[24], while the first law of the iterated logarithm was not proved until a few years

ago in [5].

We use results from the paper by Chen [8] in several places. This paper studies

moderate deviations and laws of the iterated logarithm for the joint range of several

independent random walks, that is, the cardinality of the set of points which are

each visited by each of the random walks; see (3.2). Also related is the paper by

Bass and Rosen [6], which is an almost sure invariance principle for the range. Our

results in Theorems 1.1 and 1.5 are more precise than what can be obtained using

[6]. We did not see how to derive our laws of the iterated logarithm from that

paper; moreover in that paper 2 + δ moments were required.

Acknowledgment: We would like to thank Greg Lawler and Takashi Kumagai

for helpful discussions and their interest in this paper.