Abstract
For an arbitrary transient random walk $$(S_n)_{n\ge 0}$$
(
S
n
)
n
≥
0
in $${\mathbb {Z}}^d$$
Z
d
, $$d\ge 1$$
d
≥
1
, we prove a strong law of large numbers for the spatial sum $$\sum _{x\in {\mathbb {Z}}^d}f(l(n,x))$$
∑
x
∈
Z
d
f
(
l
(
n
,
x
)
)
of a function f of the local times $$l(n,x)=\sum _{i=0}^n{\mathbb {I}}\{S_i=x\}$$
l
(
n
,
x
)
=
∑
i
=
0
n
I
{
S
i
=
x
}
. Particular cases are the number of
visited sites [first considered by Dvoretzky and Erdős (Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp 353–367, 1951)], which corresponds to the function $$f(i)={\mathbb {I}}\{i\ge 1\}$$
f
(
i
)
=
I
{
i
≥
1
}
;
$$\alpha $$
α
-fold self-intersections of the random walk [studied by Becker and König (J Theor Probab 22:365–374, 2009)], which corresponds to $$f(i)=i^\alpha $$
f
(
i
)
=
i
α
;
sites visited by the random walk exactly j times [considered by Erdős and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960) and Pitt (Proc Am Math Soc 43:195–199, 1974)], where $$f(i)={\mathbb {I}}\{i=j\}$$
f
(
i
)
=
I
{
i
=
j
}
.
Strong Law of Large Numbers for a Function of the Local Times of a Transient Random Walk in $${\mathbb {Z}}^d$$
