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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.

Approximation by Penultimate Stable Laws

de Haan L., Peng L., Pereira H. I.

Probability and Mathematical Statistics

1999

Vol. 19, Fasc. 1

105--121

In certain cases partial sums of i.i.d. random variables with finite variance are better approximated by a sequence of stable distributions with indices αn → 2 than by a normal distribution. We discuss when this happens and how much the convergencerate can be improved by using penultimate approximations. Similar results are valid for other stable distributions.