Regression conditions that characterize free-Poisson and free-Kummer distributions

Abstract

We find the asymptotic spectral distribution of random Kummer matrix. Then we formulate and prove a free analogue of HV independence property, which is known for classical Kummer and Gamma random variables and for Kummer and Wishart matrices. We also prove a related characterization of free-Kummer and free-Poisson (Marchenko–Pastur) non-commutative random variables.

The Matsumoto-Yor property in free probability via subordination and Boolean cumulants

“…Problems of this type where the Cauchy-Stieltjes transform emerges as solution to a certain quadratic equation that depends on unknown parameters, appear frequently in characterization problems in free probability. See for example already mentioned article [6] or [9] where characterization the free-GiG and free-Poisson distributions was studied. The proof of uniqueness in those two papers rely on the study of roots of ∆(z)-the discriminant of equation for G(z) and the geometric argument is given to show that ∆(z) has a desired form.…”

“…satisfying 0 < a < b. The free-Kummer distribution was defined for α > 1 in [6] as a limit of empirical spectral distribution of Kummer matrices. The above definition extends the definition from that paper for α ∈ (0, 1) and one can easily check that for α ∈ (0, 1), β ∈ R, γ > 0 we have…”

“…We note here that in [6] the parameter α was assumed to be positive but not necessarily greater than 1. One can see that the condition α ≤ 1 contradicts some assumptions made in that paper and the definition of free-Kummer distribution does not work in this case.…”

“…The Cauchy-Stieltjes transform of K(α, β, γ) for α > 1 was calculated in [6] and is given by the following formula…”

“…The existence of a limiting spectral distribution for Kummer matrices was established in [6]. It was proved that for X n ∼ K n (a n , b n , c n ) where parameters a n > n−1 2 , b n ∈ R, c n > 0 are such that 2an n → α > 1, 2bn n → β, 2an n → γ > 0 then the sequence of empirical spectral distributions µ Xn converges almost surely to what we call in this paper the free-Kummer distribution K(α, α + β, γ).…”