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Monotone convolution semigroups

100%
Hasebe T.
Studia Mathematica
|
2010
|
tom 200
|
nr 2
175-199
EN
We study how a property of a monotone convolution semigroup changes with respect to the time parameter. Especially we focus on "time-independent properties": in the classical case, there are many properties of convolution semigroups (or Lévy processes) which are determined at an instant, and moreover, such properties are often characterized by the drift term and Lévy measure. In this paper we exhibit such properties of monotone convolution semigroups; an example is the concentration of the support of a probability measure on the positive real line. Most of them are characterized by the same conditions on drift terms and Lévy measures as known in probability theory. These kinds of properties are mapped bijectively by a monotone analogue of the Bercovici-Pata bijection. Finally we compare such properties with classical, free, and Boolean cases, which will be important in an approach to unify these notions of independence.
2
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Differential independence via an associative product of infinitely many linear functionals

100%
Hasebe T.
Colloquium Mathematicum
|
2011
|
tom 124
|
nr 1
79-94
EN
We generalize the infinitesimal independence appearing in free probability of type B in two directions: to higher order derivatives and other natural independences: tensor, monotone and Boolean. Such generalized infinitesimal independences can be defined by using associative products of infinitely many linear functionals, and therefore the associated cumulants can be defined. These products can be seen as the usual natural products of linear maps with values in formal power series.
3
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Semigroups related to additive and multiplicative, free and Boolean convolutions

63%
Arizmendi O., Hasebe T.
Studia Mathematica
|
2013
|
tom 215
|
nr 2
157-185
EN
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.
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