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Domain of attraction of Gaussian probability operators in quantum limit theory

Lubnauer K., Łuczak A.

Probability and Mathematical Statistics

2010

Vol. 30, Fasc. 1

141--152

We characterise the class of probability operators belonging to the domain of attraction of Gaussian limits in the setup which is a slight generalisation of Urbanik’s scheme of noncommutative probability limit theorems.

Alternative conditions for attraction to stable vectors

de Haan L., Li D., Peng L., Iglesias-Pereira H.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 2

303--317

Relying on Geluk and de Haan [3] we derive alternative necessary and sufficient conditions for the domain of attraction of a stable distribution in Rd which are phrased entirely in terms of (joint distributions of) linear combinations of the marginals. The conditions in terms of characteristic functions should be useful for determining rates of convergence, as in de Haan and Peng [4].

Stable probability distributions and their domains of attraction : a direct approach

Geluk J. L., de Haan L.

Probability and Mathematical Statistics

2000

Vol. 20, Fasc. 1

169--188

The theory of stable probability distributions and their domains of attraction is derived in a direct way (avoiding the usual route via infinitely divisible distributions) using Fourier transforms. Regularly varying functions play an important role in the exposition.

Domains of attraction on Sturm-Liouville hypergroups of polynomial growth

Scheffler H. - P., Zeuner HM.

Journal of Applied Analysis

1999

Vol. 5, nr 2

153-170

Let (Sn : n≥ 0) be a random walk on a hypergroup (R+, *) of polynomial growth. We show that the possible limit laws of the form cn • Sn→ μ, (weakly), cn > 0, are the stable laws of the Bessel-Kingman hypergroup (wzór) for a specific α ≥ - 1/2 depending on the growth of the hypergroup. Furthermore we describe the domain of attraction (with respect to the convolution *) of these stable laws in terms of the regular variation of the Fourier transform.