## Approximation of a free Poisson process by systems of freely independent particles
Bożejko, M., da Silva, J. L., Kuna, T. and Lytvynov, E.
(2018)
It is advisable to refer to the publisher's version if you intend to cite from this work. See Guidance on citing. To link to this item DOI: 10.1142/s0219025718500200 ## Abstract/SummaryLet σ σ be a non-atomic, infinite Radon measure on ℝd, for example, dσ(x)=zdx dσ(x)=zdx where z>0 z>0 . We consider a system of freely independent particles x 1 ,…,x N x1,…,xN in a bounded set Λ⊂R d Λ⊂ℝd , where each particle x i xi has distribution 1σ(Λ) σ 1σ(Λ)σ on Λ Λ and the number of particles, N N , is random and has Poisson distribution with parameter σ(Λ) σ(Λ) . If the particles were classically independent rather than freely independent, this particle system would be the restriction to Λ Λ of the Poisson point process on R d ℝd with intensity measure σ σ . In the case of free independence, this particle system is not the restriction of the free Poisson process on R d ℝd with intensity measure σ σ . Nevertheless, we prove that this is true in an approximative sense: if bounded sets Λ (n) Λ(n) (n∈N n∈ℕ ) are such that Λ (1) ⊂Λ (2) ⊂Λ (3) ⊂⋯ Λ(1)⊂Λ(2)⊂Λ(3)⊂⋯ and ⋃ ∞ n=1 Λ (n) =R d ⋃n=1∞Λ(n)=ℝd , then the corresponding particle system in Λ (n) Λ(n) converges (as n→∞ n→∞ ) to the free Poisson process on R d ℝd with intensity measure σ σ . We also prove the following (N/V) (N/V) -limit: Let N (n) N(n) be a deterministic sequence of natural numbers such that lim n→∞ N (n) /σ(Λ (n) )=1 limn→∞N(n)/σ(Λ(n))=1 . Then the system of N (n) N(n) freely independent particles in Λ (n) Λ(n) converges (as n→∞ 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.
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