For a random walk
we introduce the following
quantities:
The
is understood as a permutation acting on
.

Proposition

(Spitzer identity)

1. We have for
:

2.
is finite a.s. iff
in which case we
have

Proof

(1) We
calculate
Note that
.
Indeed, the
says that among positions
the maximum is attained at
-th
and the
says that among positions
the maximum is attained at
.
Hence, we
continue
Note that
and
belong to
pre-
and
post-
event fields, hence, these are
independent:
The
is distributed as
:
We write the expression of interest and substitute the latest
result:
The last expression has the structure
.
One may see by concentrating at each
term that such structure is the product
.
Hence,
We calculate each term of the formula
.
Note that
.
Also, note that
.
Hence,
.
We
continue:
At this point we invoke the proposition
(
Ch.f. of entrance time
1
):
with
substituted for
and
and
obtain
We calculate the second term of the formula
similarly, using
:
We collect the
results:

Proposition

Notation

We introduce the r.v.
and
as
follows

Proposition

For each
the random vectors
and
have the same distribution.

For each
the random vectors
and
have the same distribution.

Proposition

For

Proposition

If the common distribution of stationary independent process is symmetric
then