Page 112 - Special Topic Session (STS) - Volume 3

STS518 Steen T.
widely referred to as the Bercovici-Pata bijection. It was shown in [2] that Λ is
also a homeomorphism with respect to weak convergence. This mapping
further supports the roles of the semi-circular distribution and the Marchenko-
Pastur distribution as the free analogs of the Gaussian and Poisson
distributions, respectively. Indeed, Λ maps the Gaussian distributions onto the
semi-circular ones and the Poisson distributions onto the Marchenko-Pastur
distributions. It is also noteworthy that the Cauchy distribution (see Example
2.6) is a fixed point with respect to Λ.

The Lebesgue decomposition of measures in (⊞)
From the perspective of the Lebesgue-decomposition there are fundamental
differences between the classes ℐ(∗) and ℐ(⊞). In [3] it was proved that if
ν ∈ ℐ(⊞), then ν has no continuous singular part. Furthermore, Theorem 2.5
implies that ν has at most one atom, so that ℐ(⊞) contains no
nondegenerate discrete probability laws. We mention also that it was proved
in [7] that for all sufficiently large n, the convolution power ⊞ of a (non-
degenerate) measure ν from ℐ(⊞) has no atoms. This is in contrast to the
n
fact that for a measure µ in ℐ(∗) the convolution power µ either has an atom
∗
for all n or is atom-less for all n (see e.g. [18]).
The complete picture of the Lebesgue-decomposition of a measure µ from
ℐ(⊞) is given in the following theorem which was proved only recently by
Hasebe and Sakuma (based in part on [14]).

3.7 Theorem ([11]). For a measure ν in ℐ(⊞)with free characteristic triplet
(a,ρ,η) it holds that:
(i) If a > 0 or ρ(ℝ) ∈ (1,∞], then ν is absolutely continuous (with respect to
Lebesgue measure) and has a continuous density.
(ii) If a = 0 and ρ(ℝ) = 1, then ν is absolutely continuous.
(iii) If a = 0 and ρ(ℝ) ∈ [0,1), then c ≔ log ϵ↓0 F μ −1 (iϵ)exists in ℝ, and ν({c}) =
1−ρ(ℝ). Here the function Fµ is the reciprocal Cauchy transform: Fµ =
1/Gµ.

Prominent probability laws in (⊞)
In this final subsection, we list a number a prominent probability laws (from
classical probability theory), which in recent years have been shown to belong
to ℐ(⊞).
(a) In [4] it was proved that the classical Gaussian distribution belongs to
ℐ(⊞). This (at the time rather surprising) result was obtained by a deep
complex analysis argument establishing that the free cumulant transform of
(0,1) can be extended analytically to all of ℂ (cf. Theorem 3.4). In [12] it was
−
subsequently established (based on [4]) that in fact (0,1) ∈ ℒ(⊞).

101 | I S I W S C 2 0 1 9