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On joint sum/max stability and sum/max domains of attraction

Hees K., Scheffler H.-P.

Probability and Mathematical Statistics

2018

Vol. 38, Fasc. 1

157--189

Let (Wi, Ji)iϵN be a sequence of i.i.d. [0, ∞) × R-valued random vectors. Considering the partial sum of the first component and the corresponding maximum of the second component, we are interested in the limit distributions that can be obtained under an appropriate scaling. In the case that Wi and Ji are independent, the joint distribution of the sum and the maximum is the product measure of the limit distributions of the two components. But if we allow dependence between the two components, this dependence can still appear in the limit, and we need a new theory to describe the possible limit distributions. This is achieved via harmonic analysis on semigroups, which can be utilized to characterize the scaling limit distributions and describe their domains of attraction.

Limiting behavior of weighted sums of heavy-tailed random vectors and applications

Pingyan C., Scheffler H.-P.

Probability and Mathematical Statistics

2004

Vol. 24, Fasc. 2

281--295

We present an integral test to determine the limiting behavior of weighted sums of i.i.d. Rd-valued random vectors belonging to the (generalized) domain of operator semistable attraction of some nonnormal law, and deduce a version of Chover’s law of the iterated logarithm for them. As applications, the corresponding limit results for some classical summability methods are also established.

Multivariable regular variation of functions and measures

Meerschaert M. M., Scheffler H.-P.

Journal of Applied Analysis

1999

Vol. 5, nr 1

125-146

Regular variation is an asymptotic property of functions and measures. The one variable theory is well-established, and has found numerous applications in both pure and applied mathematics. In this paper we present several new results on mul-tivariable regular variation for functions and measures.