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January 2010 The asymptotic behavior of densities related to the supremum of a stable process
R. A. Doney, M. S. Savov
Ann. Probab. 38(1): 316-326 (January 2010). DOI: 10.1214/09-AOP479

Abstract

If X is a stable process of index α∈(0, 2) whose Lévy measure has density cxα−1 on (0, ∞), and S1=sup0<t≤1Xt, it is known that P(S1>x)∽−1xα as x→∞ and P(S1x)∽−1ρ−1xαρ as x↓0. [Here ρ=P(X1>0) and A and B are known constants.] It is also known that S1 has a continuous density, m say. The main point of this note is to show that m(x)∽Ax−(α+1) as x→∞ and m(x)∽Bxαρ−1 as x↓0. Similar results are obtained for related densities.

Citation

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R. A. Doney. M. S. Savov. "The asymptotic behavior of densities related to the supremum of a stable process." Ann. Probab. 38 (1) 316 - 326, January 2010. https://doi.org/10.1214/09-AOP479

Information

Published: January 2010
First available in Project Euclid: 25 January 2010

zbMATH: 1185.60052
MathSciNet: MR2599201
Digital Object Identifier: 10.1214/09-AOP479

Subjects:
Primary: 60F15, 60J30

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 1 • January 2010
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