110 research outputs found
New characterization of two-state normal distribution
In this article we give a purely noncommutative criterion for the
characterization of two-state normal distribution. We prove that families of
two-state normal distribution can be described by relations which is similar to
the conditional expectation in free probability, but has no classical analogue.
We also show a generalization of Bozejko, Leinert and Speicher's formula
(relating moments and noncommutative cumulants).Comment: 19 pages, 2 figures, accepted for publication by Infinite Dimensional
Analysis, Quantum Probability and Related Topic
Appell polynomials and their relatives II. Boolean theory
The Appell-type polynomial family corresponding to the simplest
non-commutative derivative operator turns out to be connected with the Boolean
probability theory, the simplest of the three universal non-commutative
probability theories (the other two being free and tensor/classical
probability). The basic properties of the Boolean Appell polynomials are
described. In particular, their generating function turns out to have a
resolvent-type form, just like the generating function for the free Sheffer
polynomials. It follows that the Meixner (that is, Sheffer plus orthogonal)
polynomial classes, in the Boolean and free theory, coincide. This is true even
in the multivariate case. A number of applications of this fact are described,
to the Belinschi-Nica and Bercovici-Pata maps, conditional freeness, and the
Laha-Lukacs type characterization.
A number of properties which hold for the Meixner class in the free and
classical cases turn out to hold in general in the Boolean theory. Examples
include the behavior of the Jacobi coefficients under convolution, the
relationship between the Jacobi coefficients and cumulants, and an operator
model for cumulants. Along the way, we obtain a multivariate version of the
Stieltjes continued fraction expansion for the moment generating function of an
arbitrary state with monic orthogonal polynomials
Radial Bargmann representation for the Fock space of type B
Let be the probability and orthogonality measure for the
-Meixner-Pollaczek orthogonal polynomials, which has appeared in
\cite{BEH15} as the distribution of the -Gaussian process (the
Gaussian process of type B) over the -Fock space (the Fock space of
type B). The main purpose of this paper is to find the radial Bargmann
representation of . Our main results cover not only the
representation of -Gaussian distribution by \cite{LM95}, but also of
-Gaussian and symmetric free Meixner distributions on . In
addition, non-trivial commutation relations satisfied by -operators
are presented.Comment: 13 pages, minor changes have been mad
Semigroups of distributions with linear Jacobi parameters
We show that a convolution semigroup of measures has Jacobi parameters
polynomial in the convolution parameter if and only if the measures come
from the Meixner class. Moreover, we prove the parallel result, in a more
explicit way, for the free convolution and the free Meixner class. We then
construct the class of measures satisfying the same property for the two-state
free convolution. This class of two-state free convolution semigroups has not
been considered explicitly before. We show that it also has Meixner-type
properties. Specifically, it contains the analogs of the normal, Poisson, and
binomial distributions, has a Laha-Lukacs-type characterization, and is related
to the case of quadratic harnesses.Comment: v3: the article is merged back together with arXiv:1003.4025. A
significant revision following suggestions by the referee. 2 pdf figure
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