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2 1999

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Point regularity of p-stable density in Rd and Fisher information

Lewandowski M.

Probability and Mathematical Statistics

1999

Vol. 19, Fasc. 2

375--388

In the paper we prove that the n-th directional derivative of a p-stable density f (x) in the direction a can be estimated by [formula], where 0 < u < 1, and C depends also on geometrical properties of the Lévy measure. This inequality helps us to calculate the Fisher information of stable measures.

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.