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Abstract We extend the free convolution of Brown measures of $$R$$-diagonal elements introduced by Kösters and Tikhomirov [ 28] to fractional powers. We then show how this fractional free convolution arises naturally when studying the roots of random polynomials with independent coefficients under repeated differentiation. When the proportion of derivatives to the degree approaches one, we establish central limit theorem-type behavior and discuss stable distributions.
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Tree convolution for probability distributions with unbounded support
We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in Jekel and Liu (2020), which generalize the free, boolean, monotone, and orthogonal convolutions. In particular, for each rooted subtree T of the N-regular tree (with vertices labeled by alternating strings), we define the convolution \boxplus_T (µ1, . . . , µN) for arbitrary probability measures µ1, . . . , µN on R using a certain fixed-point equation for the Cauchy transforms. The convolution operations respect the operad structure of the tree operad from Jekel and Liu (2020). We prove a general limit theorem for iterated T -free convolution similar to Bercovici and Pata’s results in the free case (Bercovici and Pata 1999), and we deduce limit theorems for measures in the domain of attraction of each of the classical stable laws.
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- Award ID(s):
- 2002826
- PAR ID:
- 10432681
- Date Published:
- Journal Name:
- Alea
- Volume:
- 18
- Issue:
- 2
- ISSN:
- 1980-0436
- Page Range / eLocation ID:
- 1585–1623
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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