U-Statistic for Multivariate Stable Distributions

A U-statistic for the tail index of a multivariate stable random vector is given as an extension of the univariate case introduced by Fan (2006). Asymptotic normality and consistency of the proposed U-statistic for the tail index are proved theoretically. The proposed estimator is used to estimate the spectral measure. The performance of both introduced tail index and spectral measure estimators is compared with the known estimators by comprehensive simulations and real datasets.

Expected Number of Real Zeros of Lévy and Harmonizable Stable Random Polynomials

Assuming (A 0, A 1, . . . , An ) is a jointly symmetric α-stable random vector, 0 < α ≤ 2, a general formula for the expected number of real zeros of the random algebraic polynomial was obtained by Rezakhah in 1997. Rezakhah's formula is applied to the case where (A 0, . . . , An ) is formed by consecutive observations from the Lévy stable noise or from certain harmonizable stable processes, and more explicit formulas are derived for the expected number of real zeros of .