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New limit theorems related to free multiplicative convolution

100%

Sakuma N., Yoshida H.

Studia Mathematica

2013

tom 214

nr 3

251-264

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.

Symmetrization of probability measures, pushforward of order 2 and the Boolean convolution

100%

Młotkowski W., Sakuma N.

Banach Center Publications

2011

tom 96

nr 1

271-276

We study relations between the Boolean convolution and the symmetrization and the pushforward of order 2. In particular we prove that if μ₁,μ₂ are probability measures on [0,∞) then $(μ₁ ⨄ μ₂)^{s} = μ₁^{s} ⨄ μ₂^{s}$ and if ν₁,ν₂ are symmetric then $(ν₁ ⨄ ν₂)^{(2)} = ν₁^{(2)} ⨄ ν₂^{(2)}$. Finally we investigate necessary and sufficient conditions under which the latter equality holds.

Ograniczanie wyników

1 Banach Center Publications

1 Studia Mathematica

2 Sakuma N.

1 Młotkowski W.

1 Yoshida H.

1 2013

1 2011

Wyniki wyszukiwania

New limit theorems related to free multiplicative convolution

Symmetrization of probability measures, pushforward of order 2 and the Boolean convolution