Weakly Monotone Fock Space and Monotone Convolution of the Wigner Law

Vitonofrio Crismale (), Maria Elena Griseta () and Janusz Wysoczański ()
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Vitonofrio Crismale: Università degli Studi di Bari
Maria Elena Griseta: Università degli Studi di Bari
Janusz Wysoczański: Wroclaw University

Journal of Theoretical Probability, 2020, vol. 33, issue 1, 268-294

Abstract: Abstract We study the distribution (with respect to the vacuum state) of a family of partial sums $$S_m$$Sm of position operators on weakly monotone Fock space. We show that any single operator has the Wigner law, and an arbitrary family of them (with the index set linearly ordered) is a collection of monotone-independent random variables. It turns out that our problem equivalently consists in finding the m-fold monotone convolution of the semicircle law. For $$m=2$$m=2, we compute the explicit distribution. For any $$m>2$$m>2, we give the moments of the measure and show it is absolutely continuous and compactly supported on a symmetric interval whose endpoints can be found by a recurrence relation.

Keywords: Non-commutative probability; Monotone independence and convolution; Semicircle law; Generalized Catalan recurrences; 46L53; 46L54; 60B99; 05A18 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10959-018-0846-9

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