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• # Artykuł - szczegóły

## Studia Mathematica

2013 | 214 | 3 | 251-264

## New limit theorems related to free multiplicative convolution

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.

251-264

wydano
2013

### Twórcy

autor
• Department of Mathematics, Aichi University of Education, 1 Hirosawa, Igaya-cho, Kariya-shi, 448-8542 Japan
autor
• Department of Information Sciences, Ochanomizu University, 2-1-1, Otsuka, Bunkyo, Tokyo, 112-8610 Japan