Page 109 - Special Topic Session (STS) - Volume 3

STS518 Steen T.

with a = (1 − √λ) and b = (1 + √λ) .
2
2

By a standard computation one may verify that the square of a semicircular
distributed random variable has a Marchenko-Pastur distribution. This curious
fact provides a link between the free analogs of the Gaussian and Poisson
distributions which is not paralleled in the classical theory (but in accordance
with random matrix theory). This is also the case for the following theorem due
to Bercovici and Voiculescu, which illustrates some of the regularizing
properties of free convolution.

2.5 Theorem ([9]). Let µ and ν be (Borel-) probability measures on R. Then
for any atom γ for µ ⊞ ν, there exist atoms α and β for µ and ν, respectively,
such that

As corollaries of this result it follows that a free convolution µν can have at
most finitely many atoms and that a free convolution square µ ⊞ µ can have
at most one atom!

2.6 Examples. (1) Denote by C the (standard) Cauchy distribution, i.e. the
1 1
probability measure on ℝ with Lebesgue density ↦ . Then for any
1+ 2
(Borel-) probability measure µ on ℝ, it holds that ⊞ = ∗ , where the “∗”
on the right hand side denotes classical convolution. A proof of this intriguing
“folklore result” can be found e.g. in [13].
(2) In [8] Bercovici and Voiculescu established that there exist non-semicircular
1
2
probability measures µ and ν, such that ⊞ = √4 − 1 [−2,2] (). In
2
classical probability Cramér's Theorem asserts that if the convolution of two
probability measures µ and is a Gaussian distribution, then both µ and
have to be Gaussian distributions themselves. Thus Cramér's Theorem fails in
free probability.

3. Free Infinite divisibility
The classes of infinitely divisible, stable and self-decomposable probability
laws in free probability are obtained by replacing classical convolution by free
convolution in the definitions of the corresponding classical classes. By P(R)
we denote in the following the class of all Borel probability measures on R.

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