6 2. HISTORY
and that the corresponding lim inf is −1 a.s.
For 2 and 3 dimensions, the law of the iterated logarithm was proved by Bass
and Kumagai [5]. In 3 dimensions
(2.3) lim sup
n→∞
Rn ERn
(n log n log log n)1/2
=
(p2/π)

det Γ, a.s.,
where Γ is the covariance matrix of X1 and p = P(Sk = 0 for all k). The cor-
responding lim inf is the negative of this constant. These (and other laws of the
iterated logarithm) are consequences of an invariance principle. In [5] it was proved
that if the random walk is three dimensional, then there exists a one-dimensional
Brownian motion such that
(2.4)
Rn ERn
(

2p2/2π

det Γ)
Bn
log n
= O(

n(log
n)15/32).
The law of the iterated logarithm for the range in 2 dimensions was a bit
surprising:
(2.5) lim sup
n→∞
Rn ERn
n log log log n/(log n)2
= c1, a.s.
provided the random walk had bounded range and where c1 is an unidentified
constant. Among the results in the current paper is that we identify the constant,
we remove the restriction of bounded range, and we determine the corresponding
lim inf.
An almost sure invariance principle for Rn ERn in 4 or more dimensions was
proved by Hamana [15]. There the result is that there exists a one-dimensional
Brownian motion Bt such that for each λ 0
(2.6)
Rn ERn
(Var Rn)1/2
Bn =
O(n2/5+λ).
The almost sure invariance principle for random walks in 2 dimensions is more
complicated. Bass and Rosen [6] showed that if the random walk has mean 0, has
the identity as the covariance matrix, and has 2 + δ moments for some δ 0, then
for each k
(2.7) (log
n)k
Rn
n
+
k
j=1
(−1)j
log n

+ cX
−j
γj,n 0, a.s.
where Wt is a 2-dimensional Brownian motion, cX is a constant depending on the
random walk, and γj,n is the renormalized self-intersection local time of order j at
time 1 of the Brownian motion
Wt(n)
= Wnt/

n. The intuition behind this formula
is the following: at time n, if the process hits a point it has already hit before, then
n Rn increases by 1, and so does the number of self-intersections of the random
walk. If the point that is hit again has only been hit once before, then the double
self-intersections of the random walk increases by 1, but if this point has been hit a
number of times, then the double self-intersections will go up more than 1, and this
has to be compensated by subtracting off the number of triple self-intersections,
and so on.
Large deviations for the range have been considered by Donsker and Varadhan
[12], by Hamana [16], and by Hamana and Kesten [17]. These are related to
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