2 1. INTRODUCTION

A more precise statement than Theorem 1.1 is possible when the Xi have

slightly more than two moments.

Corollary 1.3. Suppose E[

|Xi|2(log+(|Xi|)) 1

2

+δ

] ∞ for some δ 0. Let

{bn} be a positive sequence satisfying bn → ∞ and log bn = o((log

n)1/2)

as n → ∞.

There are two constants C1, C2 0 independent of the choice of the sequence {bn}

such that

−C1 ≤ lim inf

n→∞

bn

−θ

log P Rn ≥ 2θπ

√

det Γ

n

(log n)2

log bn

≤ lim sup

n→∞

bn

−θ

log P Rn ≥ 2θπ

√

det Γ

n

(log n)2

log bn ≤ −C2 (1.4)

for any θ 0.

Remark 1.4. The constants C1, C2 are the same as in the statement of The-

orem 1.1. See Remark 1.2.

For bn tending to infinity faster than the rate given in Theorem 1.1, e.g.,

log bn = (log

n)2,

then we are in the realm of large deviations. For Section 2

for some references to results on large deviations of the range.

For the moderate deviations of −Rn = ERn − Rn we have the following. Let

κ(2, 2) be the smallest A such that

(1.5) f

4

≤ A∇f

1/2

2

f

1/2

2

for all f ∈ C1 with compact support. (This constant appeared in [2].)

Theorem 1.5. Suppose bn → ∞ and bn = o((log n)1/5) as n → ∞. For λ 0

lim

n→∞

1

bn

log P − Rn λ

nbn

(log n)2

=

−(2π)−2(det Γ)−1/2κ(2,

2)4 λ.

Comparing Theorems 1.1 and 1.5, we see that the upper and lower tails of

Rn are quite different. This is similar to the behavior of the distribution of the

self-intersection local time of planar Brownian motion. This is not surprising, since

LeGall, [24, Theorem 6.1], shows that Rn, properly normalized, converges in dis-

tribution to the self-intersection local time; see also (2.2).

The moderate deviations of Rn are quite similar in nature to those of −Ln,

where Ln is the number of self-intersections of the random walk Sn; see [4]. Again,

[24, Theorem 6.1] gives a partial explanation of this. However the case of the range

is much more diﬃcult than the corresponding results for intersection local times.

The latter case can be represented as a quadratic functional of the path, which is

amenable to the techniques of large deviation theory, while the range cannot be

so represented. This has necessitated the development of several new tools, see in

particular Sections 8 and 9, which we expect will have further applications in the

study of the range of random walks.

Theorem 1.1 gives rise to the following LIL for Rn.

Theorem 1.6.

(1.6) lim sup

n→∞

Rn

n log log log n/(log n)2

= 2π

√

det Γ, a.s.