Page 110 - Special Topic Session (STS) - Volume 3

STS518 Steen T.
3.1 Definition. (a) A measure µ from P(R) is ⊞-infinitely divisible if the
following condition is satisfied:
∀ ∈ ℕ∃ ∈ (ℝ): = ⊞ ⊞ … ⊞ ( terms).

The class of ⊞-infinitely divisible probability measures is denoted by ℐ(⊞).
(b) A measure µ from (ℝ) is -stable if the class of (increasing) affine
transformations

is stable under ⊞. The class of ⊞-stable probability measures is denoted by
(⊞).
(c) A measure µ from P(R) is ⊞-self-decomposable if the following condition
is satisfied:

The class of all ⊞-self-decomposable probability measures is denoted by
ℒ(⊞).
In the following we denote the classical counterparts of ℐ(⊞), ℒ(⊞) and
(⊞) by, respectively, ℐ (∗), ℒ (∗) and (∗). As in the classical case, we have
the following hierarchy of classes of probability measures (see [7] and [2]):

where (⊞) denotes the class of semi-circular distributions.
As in classical probability the main tool for studying the class ℐ(⊞) is a
Lévy -Khintchine type representation. In order to describe this in detail we first
have to introduce the free analog of the logarithm of the Fourier transform of
+
a probability measure. By ℂ (resp. ℂ ) we denote the set of complex numbers
−
with strictly positive (resp. negative) imaginary part.

3.2 Theorem & Definition ([7]). Let µ be a probability measure on ℝ, and
consider its Cauchy (or Stieltjes) transform:

Then the range of G contains an open region D in the form:
µ
µ

for suitable ϵ, δ in (0, ∞). On this region the (right) inverse G is well-defined,
−1
μ
and the free cumulant transform may subsequently be defined as
μ

The key property of the free cumulant transform is that it linearizes free
additive convolution:

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