2. HISTORY 7
estimates of the type
P(Rn ≥ ϕ(n)) or P(Rn ≤ ϕ(n))
for functions ϕ(n) that grow relative quickly. By contrast, in our paper we look at
P(Rn − ERn ≥ ϕ(n))
for functions ϕ(n) that grow not quite so quickly. The paper  determines
when X1 is in the domain of attraction of a symmetric stable process of index α.
This can be used to obtain information on P(Rn ≤ ϕ(n)). It is shown in  that
log P(Rn ≥ nx)
exists, while  examines
Related to our paper is one by Chen , which looks at moderate deviations
of the joint range Jn, the cardinality of the set of points that are each visited by
each of p independent random walks.
We will be much briefer concerning the literature of intersection local times.
There are a large number of papers, but we only mention those relevant to this pa-
per. Given two independent Brownian motions Vt, Wt in two dimensions, formally
the intersection local time is the quantity
δ0(Vs − Wu) du ds,
which is a measure of how often the two Brownian motions intersect each other;
here δ0 is the delta function. To give a meaning to this, we define the intersection
local time by
pε(Vs − Wu) ds du,
where pε is a suitable approximation to the identity, for example, the density of
a two-dimensional Brownian motion at time ε. If one now wants to define self-
intersection local time of one planar Brownian motion, one cannot simply replace
Vs by Ws. If one does, one gets an identically infinite random process. Varadhan
 proved that provided one renormalizes properly, one can get a finite limit.
There are a number of renormalizations possible. We use the following: let
(2.9) γε(t) =
pε(Ws − Wu) ds du,
(2.10) γt = lim[γε(t)
The limit exists almost surely, and is called renormalized self-intersection local time
for planar Brownian motion.
LeGall  proved that