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Weakly Monotone Fock Space and Monotone Convolution of the Wigner Law

Author

Listed:
  • Vitonofrio Crismale

    (Università degli Studi di Bari)

  • Maria Elena Griseta

    (Università degli Studi di Bari)

  • Janusz Wysoczański

    (Wroclaw University)

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.

Suggested Citation

  • Vitonofrio Crismale & Maria Elena Griseta & Janusz Wysoczański, 2020. "Weakly Monotone Fock Space and Monotone Convolution of the Wigner Law," Journal of Theoretical Probability, Springer, vol. 33(1), pages 268-294, March.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:1:d:10.1007_s10959-018-0846-9
    DOI: 10.1007/s10959-018-0846-9
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