New limit theorems related to free multiplicative convolution

• Autorzy: Noriyoshi Sakuma , Hiroaki Yoshida
• Języki publikacji: EN

Abstrakty

EN

Let ⊞, ⊠, and ⊎ be the free additive, free multiplicative, and boolean additive convolutions, respectively. For a probability measure μ on [0,∞) with finite second moment, we find a scaling limit of $(μ^{⊠ N})^{⊞ N}$ as N goes to infinity. The 𝓡-transform of its limit distribution can be represented by Lambert's W-function. From this, we deduce that the limiting distribution is freely infinitely divisible, like the lognormal distribution in the classical case. We also show a similar limit theorem by replacing free additive convolution with boolean convolution.

Kategorie tematyczne

• 46L54: Free probability and free operator algebras

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2013
• Tom: 214
• Numer: 3
• Strony: 251-264

Daty

wydano

2013

Twórcy

autor

Noriyoshi Sakuma

• Department of Mathematics, Aichi University of Education, 1 Hirosawa, Igaya-cho, Kariya-shi, 448-8542 Japan

autor

Hiroaki Yoshida

• Department of Information Sciences, Ochanomizu University, 2-1-1, Otsuka, Bunkyo, Tokyo, 112-8610 Japan

Identyfikatory

DOI

10.4064/sm214-3-4