On Free Generalized Inverse Gaussian Distributions

Abstract

We study here properties of free Generalized Inverse Gaussian distributions (fGIG) in free probability. We show that in many cases the fGIG shares similar properties with the classical GIG distribution. In particular we prove that fGIG is freely infinitely divisible, free regular and unimodal, and moreover we determine which distributions in this class are freely selfdecomposable. In the second part of the paper we prove that for free random variables X, Y where Y has a free Poisson distribution oneX +Y if and only if X has fGIG distribution for special choice of parameters. We also point out that the free GIG distribution maximizes the same free entropy functional as the classical GIG does for the classical entropy.

Homomorphisms relative to additive convolutions and max-convolutions: free, boolean and classical cases

“…The identities above somehow suggest that π corresponds to e. This correspondence has been observed in other contexts in free probability, see e.g. [30] and [21,Remark 4.5].…”

Homomorphisms relative to additive convolutions and max-convolutions: Free, boolean and classical cases

“…The fourth identity follows from the second one and b α = f α ⊠ f −1 α [3, Proposition 4.12].□ The identities above somehow suggest that π corresponds to e. This correspondence has been observed in other contexts in free probability, see e.g [30]. and[21, Remark 4.5].…”

Remarks on a Free Analogue of the Beta Prime Distribution

“…Remark 4.1. Concerning the second rationale, it should be noted that Hasebe and Szpojankowski pointed out such a correspondence between the classical and the free distributions in [21] based on the maximization problem of the entropy functionals with an external potential V . In particular, they showed the correspondence between the classical and the free generalized inverse Gaussian distributions.…”