(Uniform continuity of ch.f.)
Characteristic function of a r.v. is uniformly continuous in
.

Proposition

(Ch.f. of a sum) If
and
are independent r.v. with d.f.
and
then the r.v.
has a d.f.
and ch.f.
.

Proof

For the
part use the formula
(
Total_probability_rule
). The second
part is a direct verification.

Proposition

(Inversion of ch.f. into p.m. 1) If
is a p.m. induced by a r.v.
then for
we
have

Proof

We proceed by direct verification. We substitute the definition of
into the claim of the
proposition:
We would like to invert the order of integration using the proposition
(
Fubini theorem
). Hence, we need to verify that
the function
is
integrable over
.
We
estimate
and the
integral
is finite. Hence, the proposition (
Fubini
theorem
) is applicable and we reverse the order of
integration:
We calculate the integral with respect to
:
We expand the last integral with the help of
and note that
is an odd function of
.
Hence,
.
where the last equality holds because
is an even function of
.
The above is being passed to the limit
.
Hence, the next task is to calculate the integral of the
form
First, we calculate the
integral
The function
is sharply decaying a positive infinities for
and
.
Hence, the proposition (
Fubini theorem
) applies
and we reverse the order of
integration:
We calculate the internal integral via the repeated integration by parts:
Then
Note that if
is positive
then
If
is negative
then
If
is zero
then
Hence,

We substitute the last result into the
integral
Hence, by the proposition (
Dominated
convergence theorem
) the limit interchanges with the integral and we
recover the
result:

Proposition

(Inversion of ch.f. into p.m. 2) If
is a p.m. induced by a r.v.
then

Proposition

(Inversion of ch.f. into d.f.). Let
be a r.v. with ch.f.
and d.f.
.
Assume that
.
Then
is continuously differentiable and