×

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

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
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Alili, L. and Chaumont, L. (2000). A new fluctuation identity for Lévy processes and some applications. Preprint 584, Univ. Paris VI. · Zbl 1063.60062
[2] Alili, L. and Chaumont, L. (2001). A new fluctuation identity for Lévy processes and some applications. Bernoulli 7 557-569. · Zbl 1003.60045
[3] Alili, L. and Kyprianou, A. E. (2007). Reformulations of some fluctuation identities for Lévy processes.
[4] Alili, L. and Doney, R. A. (1999). Wiener-Hopf factorization revisited and some applications. Stochastics Stochastics Rep. 66 87-102. · Zbl 0928.60067
[5] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121 . Cambridge Univ. Press, Cambridge. · Zbl 0861.60003
[6] Bernyk, V., Dalang, R. C. and Peskir, G. (2008). The law of the supremum of a stable Lévy process with no negative jumps. Ann. Probab. 36 1777-1789. · Zbl 1185.60051
[7] Bingham, N. H. (1973). Maxima of sums of random variables and suprema of stable processes. Z. Wahrsch. Verw. Gebiete 26 273-296. · Zbl 0238.60036
[8] Chaumont, L. (1994). Sur certains processus de Lévy conditionnés à rester positifs. Stochastics Stochastics Rep. 47 1-20. · Zbl 0827.60064
[9] Chaumont, L. and Doney, R. A. (2005). On Lévy processes conditioned to stay positive. Electron. J. Probab. 10 948-961. · Zbl 1109.60039
[10] Chaumont, L. and Doney, R. A. (2008). On Lévy processes conditioned to stay positive: Correction. Electron. J. Probab. 13 1-4. · Zbl 1189.60097
[11] Doney, R. A. (2008). A note on the supremum of a stable process. Stochastics 80 151-155. · Zbl 1139.60022
[12] Vatutin, V. A. and Wachtel, V. (2009). Local probabilities for random walks conditioned to stay positive. Probab. Theory Related Fields 143 177-217. · Zbl 1158.60014
[13] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68 . Cambridge Univ. Press, Cambridge. · Zbl 0973.60001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.