Page 111 - Special Topic Session (STS) - Volume 3

STS518 Steen T.
3.3 Theorem ([21],[15],[7]). For any (Borel-) probability measures µ , µ on ℝ
1
2
we have that

−
for all z in a region of ℂ where all three free cumulant transforms are defined.
We can now state a Lévy-Khintchine type representation for the free
cumulant transform of a free infinitely divisible probability law:

3.4 Theorem ([7]). For a measure µ in P(R) the following conditions are
equivalent:
(i) µ ∈ ℐ(⊞).
(ii) may be extended to an analytic function : ℂ → ℂ.
−
µ
µ
(iii) There exist unique a in [0, ∞), η in ℝ and a Lévy measure ρ on ℝ, such
that

The triplet (a, ρ, η) appearing in (iii) of Theorem 3.4 is called the free
characteristic triplet for µ. In terms of the characteristic triplet, we have the
following characterizations of the classes of stable and self-decomposable
probability laws in free probability:

3.5 Theorem ([7], [2]). For µ in ℐ(⊞) with free characteristic triplet (a, ρ, η)
it holds that:
(i) μ ∈ (⊞)Γ(⊞), if and only if a = 0, and ρ has the form:

for suitable constants c c in [0,∞) and α in (0,2).
+ −
k(t)
(ii) μ ∈ ℒ(⊞), if and only if ρ has the form: ρ(dt) = dt, where k is
|t|
increasing on (−∞, 0) and decreasing on (0, ∞).

3.6 The Bercovici-Pata bijection. The theorem above demonstrates a
complete analogy to the characterizations of the classical stable and
selfdecomposable distributions in terms of the classical Lévy-Khintchine
representation. This is no coincidence. In fact the mapping that maps a
measure in ℐ(∗) with characteristic triplet (, , ) onto the measure in
ℐ(⊞) with free characteristic triplet (, , ) is (obviously) bijection, but it
also preserves scaling of measures and satisfies that ( ) = (for all in ℝ)

and that (µ ∗ µ ) = (µ ) ⊞ (µ ) for all µ , µ inℐ(∗)). These properties
2
1
2
1
2
1
immediately imply that ((∗)) = (⊞) and (ℒ(∗)) = ℒ(⊞), and hence
the characterizations in Theorem 3.5 follow readily from the corresponding
classical results (see e.g. [18]). The mapping Λ was introduced in [6] and is
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