(Wiener-Hopf technique) Let
where the functions
are Fourier transforms of measures with support on
(or
),
the functions
are Fourier transforms of measures with support on
(or
respectively).

Suppose that there exists a
such that the power series
converge for
and
then

Proof

The equality
may be written as
We look at terms
for every
:

According to the proposition (
Inversion
of ch.f. into p.m.
1
),
where the
is the measure that produces the Fourier transform
.
There is a similar statement for
and
.
We write the formula
in the
form
and note that the LHS and RHS are produced from measures with disjoint
supports. Hence, we must
have

With such result in place we proceed to the equation
and
find
and so
fourth:

Proposition

(Ch.f. of entrance time 1) Let
be the time of first entrance into the interval
with
being one of the following:
,
,
,
:

Then

Proof

We prove the statement for the case
.

For an
,
and
being the ch.f. function of
we
write
Using the proposition (
Ch.f. of a sum
) we
substitute
:
and separate the sum into the
pre-
and
post-
parts
We calculate the first
term:
We interchange the order of summation
:
where the
is the Fourier transform of the measure
having support outside of
.
Indeed,
implies that
.

We calculate the second
term:
We make the change of summation index
:
By independence of
pre-
and
post-
fields,
The
is distributed like
:
where the
:
is the Fourier transform of the measure
having support in
.

We put our results
together:
We transform the
as follows. The Taylor decomposition of
at
reads:
Therefore,
The statement now follows from the proposition
(
Wiener-Hopf technique
) and the formula
.

Until end of this section, the set
and the r.v.
are defined as in the proposition (
Ch.f. of
entrance time 1
).

Proposition

We have
and
iff
in which case

Proposition

Suppose that
and at least one of
is finite,
then

Proposition

If
and
then

Proposition

Suppose that
and at least one of
is finite. Let
,
.