• Bernoulli
  • Volume 21, Number 4 (2015), 2139-2156.

Poisson convergence on the free Poisson algebra

Solesne Bourguin

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Based on recent findings by Bourguin and Peccati, we give a fourth moment type condition for an element of a free Poisson chaos of arbitrary order to converge to a free (centered) Poisson distribution. We also show that free Poisson chaos of order strictly greater than one do not contain any non-zero free Poisson random variables. We are also able to give a sufficient and necessary condition for an element of the first free Poisson chaos to have a free Poisson distribution. Finally, depending on the parity of the considered free Poisson chaos, we provide a general counterexample to the naive universality of the semicircular Wigner chaos established by Deya and Nourdin as well as a transfer principle between the Wigner and the free Poisson chaos.

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Bernoulli Volume 21, Number 4 (2015), 2139-2156.

Received: December 2013
Revised: April 2014
First available in Project Euclid: 5 August 2015

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chaos structure combinatorics of free Poisson random measures contractions diagram formulae fourth moment theorem free Poisson distribution free probability multiplication formula


Bourguin, Solesne. Poisson convergence on the free Poisson algebra. Bernoulli 21 (2015), no. 4, 2139--2156. doi:10.3150/14-BEJ638.

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