Necessary conditions for the existence of conditional moments of stable random variables
Let (X1, X2) be a symmetric [alpha]-stable random vector with 0 < [alpha] < 2. Its distribution is characterized by a finite measure [GH] on the unit circle called the spectral measure. It is known that if [GH] satisfies some integrability condition then the conditional moment E[X2pX1 = x] can exist for [alpha] [less-than-or-equals, slant] p < 2[alpha] + 1. The paper shows that this sufficient condition is also necessary in the cases [alpha] < p < 2[alpha] + 1 if either 0 < [alpha] [less-than-or-equals, slant] 1/2 or 1 < [alpha] [less-than-or-equals, slant] 3/2, [alpha] < p [less-than-or-equals, slant] 2 if 1/2 < [alpha] [less-than-or-equals, slant] 1 and [alpha] < p [less-than-or-equals, slant] 4 if 3/2 < [alpha] < 2. It also provides a sufficient and necessary condition for the existence of E[X2[alpha]X1 = x] (i.e. P = [alpha]) for 0 < [alpha] < 2.
Year of publication: |
1995
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Authors: | Cioczek-Georges, Renata ; Taqqu, Murad S. |
Published in: |
Stochastic Processes and their Applications. - Elsevier, ISSN 0304-4149. - Vol. 56.1995, 2, p. 233-246
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Publisher: |
Elsevier |
Keywords: | Stable distributions Bivariate stable distributions Conditional moments Regression |
Saved in:
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