## The asymptotic behavior of densities related to the supremum of a stable process
Doney, R.A. and Savov, M.S.
(2010)
Full text not archived in this repository. It is advisable to refer to the publisher's version if you intend to cite from this work. See Guidance on citing. To link to this item DOI: 10.1214/09-AOP479 ## Abstract/SummaryIf X is a stable process of index α∈(0, 2) whose Lévy measure has density cx−α−1 on (0, ∞), and S1=sup0<t≤1Xt, it is known that P(S1>x)∽Aα−1x−α as x→∞ and P(S1≤x)∽Bα−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.
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