Semigroups related to additive and multiplicative, free and Boolean convolutions

Octavio Arizmendi, and Takahiro Hasebe. "Semigroups related to additive and multiplicative, free and Boolean convolutions." Studia Mathematica 215.2 (2013): 157-185. <http://eudml.org/doc/285384>.

@article{OctavioArizmendi2013,
abstract = {Belinschi and Nica introduced a composition semigroup of maps on the set of probability measures. Using this semigroup, they introduced a free divisibility indicator, from which one can know quantitatively if a measure is freely infinitely divisible or not. In the first half of the paper, we further investigate this indicator: we calculate how the indicator changes with respect to free and Boolean powers; we prove that free and Boolean 1/2-stable laws have free divisibility indicators equal to infinity; we derive an upper bound of the indicator in terms of Jacobi parameters. This upper bound is achieved only by free Meixner distributions. We also prove Bożejko’s conjecture that the Boolean powers $μ^\{⊎t\}$, t ∈ [0,1], of a probability measure μ are freely infinitely divisible if the measure μ is freely infinitely divisible. In the other half of the paper, we introduce an analogous composition semigroup for multiplicative convolutions and define free divisibility indicators for these convolutions. Moreover, we prove that a probability measure on the unit circle is freely infinitely divisible relative to the free multiplicative convolution if and only if the indicator is not less than one. We also prove how the multiplicative divisibility indicator changes under free and Boolean powers and then we establish the multiplicative analogue of Bożejko’s conjecture. We include an appendix, where the Cauchy distributions and point measures are shown to be the only fixed points of the Boolean-to-free Bercovici-Pata bijection.},
author = {Octavio Arizmendi, Takahiro Hasebe},
journal = {Studia Mathematica},
keywords = {free convolution; Boolean convolution; free divisibility indicator; composition semigroup; freely infinitely divisible},
language = {eng},
number = {2},
pages = {157-185},
title = {Semigroups related to additive and multiplicative, free and Boolean convolutions},
url = {http://eudml.org/doc/285384},
volume = {215},
year = {2013},
}

TY - JOUR
AU - Octavio Arizmendi
AU - Takahiro Hasebe
TI - Semigroups related to additive and multiplicative, free and Boolean convolutions
JO - Studia Mathematica
PY - 2013
VL - 215
IS - 2
SP - 157
EP - 185
AB - Belinschi and Nica introduced a composition semigroup of maps on the set of probability measures. Using this semigroup, they introduced a free divisibility indicator, from which one can know quantitatively if a measure is freely infinitely divisible or not. In the first half of the paper, we further investigate this indicator: we calculate how the indicator changes with respect to free and Boolean powers; we prove that free and Boolean 1/2-stable laws have free divisibility indicators equal to infinity; we derive an upper bound of the indicator in terms of Jacobi parameters. This upper bound is achieved only by free Meixner distributions. We also prove Bożejko’s conjecture that the Boolean powers $μ^{⊎t}$, t ∈ [0,1], of a probability measure μ are freely infinitely divisible if the measure μ is freely infinitely divisible. In the other half of the paper, we introduce an analogous composition semigroup for multiplicative convolutions and define free divisibility indicators for these convolutions. Moreover, we prove that a probability measure on the unit circle is freely infinitely divisible relative to the free multiplicative convolution if and only if the indicator is not less than one. We also prove how the multiplicative divisibility indicator changes under free and Boolean powers and then we establish the multiplicative analogue of Bożejko’s conjecture. We include an appendix, where the Cauchy distributions and point measures are shown to be the only fixed points of the Boolean-to-free Bercovici-Pata bijection.
LA - eng
KW - free convolution; Boolean convolution; free divisibility indicator; composition semigroup; freely infinitely divisible
UR - http://eudml.org/doc/285384
ER -