(Convergence of p.m. and ch.f. 1) Let
be a family of p.m. on
and
is the corresponding family of ch.f.

If
vaguely then
uniformly in every finite interval and
is equicontinuous.

Proof

If
vaguely then the pointwise convergence of
follows from the proposition
(
Vague convergence as a
weak convergence 2
). To prove the equicontinuity we consider the
difference
for small
.
Since
are p.m. the limit
is also a p.m. Fix an
.
There exists a positive
from the continuity set of
such that
.
By vague convergence there exists an
such that
we also have
.
Therefore, we
continue
To estimate the first integral we note that the
is a small parameter and
is bounded by
.
Hence, it follows from Taylor decomposition of
around
that there is an
and a constant
such that
and
we have
.
Hence, we
continue
This proves equicontinuity. Since any ch.f. is bounded
the uniform convergence on any finite interval follows from the proposition
(
Arzela-Ascoli compactness
criterion
).

Proposition

(P.m. vs ch.f. estimate) Let
and
be related p.m. and ch.f. Then for any
we
have

Proof

We use the technique of the proof of the proposition
(
Inversion of ch.f. into p.m. 1
).
For this reason the verbose justification of some steps is
omitted.
Fix a constant
.
For
we have
.
For
we have
.
We
continue
Set
then

Proposition

(Convergence of p.m. and ch.f. 2) Let
be a family of p.m. on
and
be the corresponding family of ch.f. Assume

A.
pointwise everywhere on
.

B.
is continuous at
.

Then we have

a.
vaguely and
is a p.m.

b.
is a ch.f. of
.

Proof

According to the proposition (
Vague
precompactness of s.p.m.
), there is a subsequence
that converges vaguely to some s.p.m.
.
We proceed to show that
is a p.m.

Fix a large constant
such that
are continuity points of
.
We
have
By the proposition
(
Equivalent
definitions of vague
convergence
)
We seek to apply the proposition (
P.m. vs ch.f.
estimate
) to the above expression. Set
.
We
estimate
Fix
.
It follows from
and continuity of
at
that
such that
With
so fixed we use the convergence of
to
,
boundedness
and the proposition (
Dominated
convergence theorem
) to state that
such that
Hence,
We apply the proposition (
P.m. vs ch.f.
estimate
):
Therefore
for arbitrary
.
We proved that the
is a p.m.

Let
is ch.f. of
.
Then, by assumption (A) and the proposition
(
Convergence of p.m. and ch.f. 1
),
.
Therefore, every limit point
considered above has the same ch.f. Hence, by the proposition
(
Inversion of ch.f. into p.m. 1
)
every limit point of
is the same. Hence,
converges vaguely into a p.m.