THE ASYMPTOTIC BEHAVIOR OF DENSITIES RELATED TO THE SUPREMUM OF A STABLE PROCESS

If X is a stable process of index α ∈ (0, 2) whose Lévy measure has density cx −α−1 on (0, ∞), and S 1 = sup 0≺t≤1 X t , it is known that P (S 1 > x) ∼ Aα −1 x −α as x → ∞ and P(S 1 ≤ x) ∼ Bα −1 ρ −1 x αρ as x ↓ 0. [Here ρ = P(X 1 > 0) and A and B are known constants.] It is also known that S...

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Bibliographic Details
Published in:The Annals of probability 2010-01, Vol.38 (1), p.316-326
Main Authors: Doney, R. A., Savov, M. S.
Format: Article
Language:English
Subjects:
60F15
60J30
asymptotic behavior
Asymptotic methods
Density
Exact sciences and technology
General topics
Infinity
Mathematical analysis
Mathematics
passage time density
Probability and statistics
Probability theory and stochastic processes
Random walk
Sciences and techniques of general use
stable meander
Stable process
Stochastic processes
Studies
supremum
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Summary:If X is a stable process of index α ∈ (0, 2) whose Lévy measure has density cx −α−1 on (0, ∞), and S 1 = sup 0≺t≤1 X t , it is known that P (S 1 > x) ∼ Aα −1 x −α as x → ∞ and P(S 1 ≤ x) ∼ Bα −1 ρ −1 x αρ as x ↓ 0. [Here ρ = P(X 1 > 0) and A and B are known constants.] It is also known that S 1 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) ∼ Bx αρ−1 as x ↓ 0. Similar results are obtained for related densities.
ISSN:0091-1798
2168-894X
DOI:10.1214/09-aop479