How do conditional moments of stable vectors depend on the spectral measure?

AbstractLet (X1,X2) be an α-stable random vector with 0 < α < 2, not necessarily symmetric. Its distribution is characterized by a finite measure Γ on the unit circle called the spectral measure. It is known that if Γ satisfies some integrability condition then the conditional moment E[∥X2∥p∥X1] can exist for some values of p greater than α. This paper provides a sufficient condition on Γ for the existence of the conditional moment E[∥X2∥p∥X1] involving the maximal range of possible p's, namely p < 2α + 1

Stable fractal sums of pulses: the cylindrical case

A class of α-stable, 0\textlessα\textless2, processes is obtained as a sum of ’up-and-down’ pulses determined by an appropriate Poisson random measure. Processes are H-self-affine (also frequently called ’self-similar’) with H\textless1/α and have stationary increments. Their two-dimensional dependence structure resembles that of the fractional Brownian motion (for H\textless1/2), but their sample paths are highly irregular (nowhere bounded with probability 1). Generalizations using different shapes of pulses are also discussed

Does asymptotic linearity of the regression extend to stable domains of attraction?

C. D. Hardin, Jr., G. Samorodnitsky, and M. S. Taqqu (1991,Ann. Appl. Probab. 1 582-612) have shown that the regression E[Y X = x] is typically asymptotically linear when (X, Y) is an [alpha]-stable random vector with [alpha]stable distributions bivariate stable distributions domain of attraction conditional moments regression nonlinear regression

How do conditional moments of stable vectors depend on the spectral measure?

Let (X1,X2) be an [alpha]-stable random vector with 0Stable distributions Stable random vectors Symmetric [alpha]-stable Conditional moments

Necessary conditions for the existence of conditional moments of stable random variables

Let (X1, X2) be a symmetric [alpha]-stable random vector with 0Stable distributions Bivariate stable distributions Conditional moments Regression