Anna Dorota Krystek

INFINITE DIVISIBILITY FOR THE CONDITIONALLY FREE CONVOLUTION

Infinite divisibility for the free additive convolution was studied in Ref. 20. A complete characterization of -infinitely divisible distributions was given, and it was explained in Ref. 21 that this characterization is an analogue of the classical Levy–Khintchine characterization. In fact, the analogue of the Gaussian distribution appeared even earlier, when the central limit theorem for free additive convolution was proven in Ref. 19. In this paper we define the notion of -infinitely divisibility and give the description of infinitely divisible compactly supported probability measures relative to the conditionally free convolution. We also show that the Levy–Khintchine measures associated with a -infinitely divisible distribution μ can be calculated, as in the classical or free case, as a weak limit of measures related with the convolution semigroup generated by (μ, φ) for -infinitely divisible.

ASSOCIATIVE CONVOLUTIONS ARISING FROM CONDITIONALLY FREE CONVOLUTION

We define two families of deformations of probability measures depending on the second free cumulants and the corresponding new associative convolutions arising from the conditionally free convolution. These deformations do not commute with dilation of measures, which means that the limit theorems cannot be obtained as a direct application of the theorems for the conditionally free case. We calculate the general form of the central and Poisson limit theorems. We also find the explicit form for three important examples.

THE COMBINATORICS OF THE r-FREE CONVOLUTION

In this paper we shall give combinatorial remarks on the r-free convolution. In particular, we shall introduce the set partition statistic on non-crossing partitions, which gives the r-free deformed moment-cumulant formula. We shall also give the probability measure of the r-free Poisson law and its moments, exactly.

CONVOLUTION AND CENTRAL LIMIT THEOREM ARISING FROM ADDITION OF FIELD OPERATORS IN ONE-MODE TYPE INTERACTING FOCK SPACES

In Ref. 2 the authors introduced field operators in one-mode type Interacting Fock Spaces whose spectral measures have common symmetric Jacobi recurrence coefficients but differ in the nonsymmetric ones. We show that the convolution of measures arising from addition of such field operators is the universal convolution of Accardi and Bozejko. We also present the associated central limit theorem in a more general form than in Ref. 2 and give it a proof based on the properties of the convolution.

Interpolations of Bargmann Type Measures

Abstract In this paper, we shall discuss Bargmann type measures on C for several classes of probability measures on R. The unified interpolation expressions include not only the classical Bargmann measure and its q-deformation, but also their t-deformations and dilations. As a special case, we get conditions on existence and an explicit form of the Bargmann representation for the free Meixner family of probability measures.

Conditionally free semi-stable distributions

We define a notion of semi–stability in the conditionally free probability and explain that the semi–stable measures are infinitely divisible. We also show that in the conditionally free probability stable measures are semi–stable, and that semi–stability for all r implies stability.