The arithmetic of distributions in free probability theory

By Gennadiy P. Chistyakov, Friedrich Götze. 2005

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We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and Schur functions. We consider the set of probability distributions as a semigroup M equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of M contains either indecomposable (“prime”) factors or it belongs to a class, say I0, of distributions without indecomposable factors. In contrast to the classical convolution semigroup, in the free additive and multiplicative convolution semigroups the class I0 consists of units (i.e. Dirac measures) only. Furthermore we show that the set of indecomposable elements is dense in M.

Subfields: Finance, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7178153a8c446955cba6c08d1f42552d10d5f13d

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Citations in the corpus

The normal distribution is -infinitely divisible (2009, background): “This subject is not new (implicit results on the arithmetic of free convolutions can be found in many works), but it is the rather recent preprint [20] that has first addressed the problem of the decomposability of measures in “free convolution factors” in an explicit and systematic way.”
Free Probability and Random Matrices (2014, background): “It was noted, and for the ﬁrst time explicitly formulated in [15], that the sub-ordination property is equivalent to the R -transform approach, but has better analytic properties.”
Lectures on the Combinatorics of Free Probability (2006, methodology): “Similar to the additive free convolution, there exists an extensive literature around , see [9, 19] for details and further references.”
Edge universality for deformed Wigner matrices (2014, background): “11), the free convolution measure ω1 ⊞ ω2 can be described in terms of a set of functional equations for the Stieltjes transforms; see [10, 3].”
Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem (2013, methodology): “, [4]) that using a subordination approach to free convolution (which, as also noted – and for the first time explicitly formulated – in [14], is equivalent to the R-transform approach, but has better analytic properties) is more suited for analytic questions.”
The Lebesgue decomposition of the free additive convolution of two probability distributions (2006, background): “The same argument shows the required statement for ω2 This proves the second statement of the proposition.”
Limit theorems in free probability theory. I (2006, methodology): “[11] Chistyakov, G. P. and Götze, F. Limit theorems in free probability theory.”