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 log log n/(log n)2

det Γ κ(2,
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.
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