Approximation of a free Poisson process by systems of freely independent particles

Abstract

Let σ be a non-atomic, infinite Radon measure on R d , for example, dσ(x) = z dx where z > 0. We consider a system of freely independent particles x 1 , . . . , x N in a bounded set Λ ⊂ R d , where each particle x i has distribution 1 σ(Λ) σ on Λ and the number of particles, N , is random and has Poisson distribution with parameter σ(Λ). If the particles were classically independent rather than freely independent, this particle system would be the restriction to Λ of the Poisson point process on R d with intensity measure σ. In the case of free independence, this particle system is not the restriction of the free Poisson process on R d with intensity measure σ. Nevertheless, we prove that this is true in an approximative sense: if bounded sets Λ (n) (n ∈ N) are such that Λ (1) ⊂ Λ (2) ⊂ Λ (3) ⊂ · · · and ∞ n=1 Λ (n) = R d , then the corresponding particle system in Λ (n) converges (as n → ∞) to the free Poisson process on R d with intensity measure σ. We also prove the following N/V -limit: Let N (n) be a determinstic sequence of natural numbers such that lim n→∞ N (n) /σ(Λ (n) ) = 1. Then the system of N (n) freely independent particles in Λ (n) converges (as n → ∞) to the free Poisson process. We finally extend these results to the case of a free Lévy white noise (in particular, a free Lévy process) without free Gaussian part.

Noncommutative analogues of the Law of Small Numbers for random variables indexed by elements of positive symmetric cones

“…Many more classical properties have been studied in the noncommutative context (cf. Arizmendi and Hasebe, 2013;Bożejko et al, 2018;Lenczewski and Sałapata, 2006;Liu, 2018). Moreover, several noncommutative results have been extended to operator valued distributions (cf.…”