The asymptotic behavior of densities related to the supremum of a stable process. (English) Zbl 1185.60052

Doney, R. A.; Savov, M. S.

Ann. Probab. 38, No. 1, 316-326 (2010).

Summary: If \(X\) is a stable process of index \(\alpha\in(0, 2)\) whose Lévy measure has density \(cx^{-\alpha-1}\) on \((0,\infty)\), and \(S_1=\sup_{0<t\leq 1}X_t\), it is known that \(P(S_1>x)\sim A\alpha-1x^{-\alpha}\) as \(x\to\infty\) and \(P(S_1\leq x)\sim B\alpha^{-1}\rho^{-1}x^{\alpha\rho}\) as \(x\downarrow 0\). [Here \(\rho=P(X_1>0)\) 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 \(m(x)\sim Ax^{-(\alpha+1)}\) as \(x\to\infty\) and \(m(x)\sim Bx^{\alpha\rho-1}\) as \(x\downarrow 0\). Similar results are obtained for related densities.

MSC:

60G52	Stable stochastic processes
60F15	Strong limit theorems
60G70	Extreme value theory; extremal stochastic processes
60E99	Distribution theory

Keywords:

stable process; stable meander; supremum; passage time density; asymptotic behavior

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