The Lebesgue decomposition of the free additive convolution of two probability distributions

By Serban Teodor Belinschi. 2006

79 citations recorded by the source corpus.

We prove that the free additive convolution of two Borel probability measures supported on the real line can have a component that is singular continuous with respect to the Lebesgue measure on $${\mathbb{R}}$$ only if one of the two measures is a point mass. The density of the absolutely continuous part with respect to the Lebesgue measure is shown to be analytic wherever positive and finite. The atoms of the free additive convolution of Borel probability measures on the real line have been described by Bercovici and Voiculescu in a previous paper.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7d4b40821ff81f0873c195509407abfff6e6d875

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In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Rectangular random matrices, related convolution (2005, methodology): “−, Gτ(z) = (4 −2z2)(λ−1) +2s(z) 2λz(4 −z2) where s(z) is analytic on C− and satisﬁes s(z)2 = (z2 − 2)2 − κ2. In order to compute τ, we shall use the following result (for reference, see lemma 2.17 of [B06]): for Lebesgue-almost all real number x, the limit, as z= u+ iv∈ C− tends to xin such a way th”