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# Nonlinear Regression of Stable Random Variables

The Annals of Applied Probability
Vol. 1, No. 4 (Nov., 1991), pp. 582-612
Stable URL: http://www.jstor.org/stable/2959708
Page Count: 31
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## Abstract

Let (X1, X2) be an α-stable random vector, not necessarily symmetric, with $0 < \alpha < 2$. This article investigates the regression E(X2 ∣ X1 = x) for all values of α. We give conditions for the existence of the conditional moment E(|X2|p|X1 = x) when p ≥ α, and we obtain an explicit form of the regression E(X2 ∣ X1 = 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 x → ± ∞. The behavior of the regression functions is also illustrated graphically.

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