Asymptotics of supremum distribution of a Gaussian process over a Weibullian time. (English) Zbl 1284.60074

Arendarczyk, Marek; Dębicki, Krzysztof

Bernoulli 17, No. 1, 194-210 (2011).

Summary: Let {\(X(t) : t\in [0, \infty)\)} be a centered Gaussian process with stationary increments and variance function \(\sigma _{X}^{2}(t)\). We study the exact asymptotics of \(\operatorname{P}(\text{sup}_{t\in [0, T]}X(t)>u)\) as \(u\rightarrow \infty \), where \(T\) is an independent of \({X(t)}\) non-negative Weibullian random variable. As an illustration, we work out the asymptotics of the supremum distribution of fractional Laplace motion.

MSC:

60G05	Foundations of stochastic processes
60G22	Fractional processes, including fractional Brownian motion
60G70	Extreme value theory; extremal stochastic processes

Keywords:

exact asymptotics; fractional Laplace motion; Gaussian process

References:

[1]	Abundo, M. (2008). Some remarks on the maximum of a one-dimensional diffusion process. Probab. Math. Statist. 28 107-120. · Zbl 1136.60356
[2]	Adler, R.J. (1990). An introduction to continuity, extrema, and related topics for general Gaussian processes. In Inst. Math. Statist. Lecture Notes Monograph Series 12 . Hayward, CA: Inst. Math. Statist. · Zbl 0747.60039
[3]	Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular Variation . Cambridge: Cambridge Univ. Press. · Zbl 0617.26001
[4]	Borst, S.C., De\ogonek bicki, K. and Zwart, A.P. (2004). The supremum of a Gaussian process over a random interval. Statist. Probab. Lett. 68 221-234. · Zbl 1072.60032
[5]	Fedoryuk, M. (1977). The Saddle-Point Method . Moscow: Nauka. · Zbl 0463.41020
[6]	Kozubowski, T.J., Meerschaert, M.M., Molz, F.J. and Lu, S. (2004). Fractional Laplace model for hydraulic conductivity. Geophys. Res. Lett. 31 1-4. L08501.
[7]	Kozubowski, T.J., Meerschaert, M.M. and Podgórski, K. (2006). Fractional Laplace motion. Adv. in Appl. Probab. 38 451-464. · Zbl 1100.60017 · doi:10.1239/aap/1151337079
[8]	Piterbarg, V.I. and Prisyazhn’uk, V. (1978). Asymptotic behaviour of the probability of a large excursion for a non-stationary Gaussian process. Theory Probab. Math. Statist. 18 121-133. · Zbl 0407.60016
[9]	Piterbarg, V.I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. Translations of Mathematical Monographs 148 . Providence, RI: Amer. Math. Soc. · Zbl 0841.60024
[10]	Talagrand, M. (1988). Small tails for the supremum of Gaussian process. Ann. Inst. H. Poincare Probab. Statist. 24 307-315. · Zbl 0641.60044
[11]	Zwart, A.P., Borst, S.C. and De\ogonek bicki, K. (2005). Subexponential asymptotics of hybrid fluid and ruin models. Ann. Appl. Probab. 15 500-517. · Zbl 1079.60037 · doi:10.1214/105051604000000648