Infinitely divisible distributions and Levy-Khintchine formula.

efinition

(Characteristic exponent of a
p.m.) Let
be a probability measure on
.
The characteristic exponent
is defined by the relationship

Definition

(Infinitely divisible p.m.) The
probability measure
on
is "infinitely divisible" if for any integer
there exists a p.m.
such
that

Proposition

(Levy-Khintchine formula 1) A function
is a characteristic function of an infinitely divisible p.m. iff there are
,
positive semi-definite matrix
matrix
and a measure
on
with
such
that
for any
.

Equivalent formulation of the above proposition may be obtained if the cut
off function
is replaced with
.
This way we have a smooth function with equivalent behavior at
and
.

Proposition

(Levy-Khintchine formula 2) A function
is a characteristic function of an infinitely divisible p.m. iff there are
,
positive semi-definite matrix
matrix
and a measure
on
with
such
that
for any
.