1,678 research outputs found
LĂ©vy measures of infinitely divisible random vectors and Slepian inequalities
We study Slepian inequalities for general non-Gaussian infinitely divisible random vectors. Conditions for such inequalities are expressed in terms of the corresponding Levy measures of these vectors. These conditions are shown to be nearly best possible, and for a large subfamily of infinitely divisible random vectors these conditions are necessary and sufficient for Slepian inequalities. As an application we consider symmetric αα\textbackslashalpha-stable Ornstein-Uhlenbeck processes and a family of infinitely divisible random vectors introduced by Brown and Rinott
(1/α)-Self similar α-stable processes with stationary increments
Originally published as a technical report no. 892, February 1990 for Cornell University Operations Research and Industrial Engineering. Available online: http://hdl.handle.net/1813/8775In this note we settle a question posed by Kasahara, Maejima, and Vervaat. We show that the α-stable Lévy motion is the only α-stable process with stationary increments if 0 < α < 1. We also introduce new classes of α-stable processes with stationary increments for 1 < α < 2.https://www.sciencedirect.com/science/article/pii/0047259X9090031C?via=ihubhttps://www.sciencedirect.com/science/article/pii/0047259X9090031C?via=ihubAccepted mansucrip
Sample path properties of stochastic processes represented as multiple stable integrals
Originally published as a technical report no. 871, October 1989 for Cornell University Operations Research and Industrial Engineering. Available online: http://hdl.handle.net/1813/8754This paper studies the sample path properties of stochastic processes represented by multiple symmetric α-stable integrals. It relates the âsmoothnessâ of the sample paths to the âsmoothnessâ of the (non-random) integrand. It also contains results about the behavior of the distribution of suprema and zero-one laws
Nonlinear regression of stable random variables
Let (X1,X2) be an α-stable random vector, not necessarily symmetric, with 0<α<2. This article investigates the regression E(X2âŁX1=x) for all values of α. We give conditions for the existence of the conditional moment E(|X2|p|X1=x) when pâ„α, and we obtain an explicit form of the regression E(X2âŁX1=x) as a function of x. Although this regression is, in general, not linear, it can be linear even when the vector (X1,X2) is skewed. We give a necessary and sufficient condition for linearity and characterize the asymptotic behavior of the regression as xâ±â. The behavior of the regression functions is also illustrated graphically
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