×

Weak convergence of subordinators to extremal processes. (English) Zbl 1319.60117

Summary: For certain subordinators \((X_t)_{t\geq 0}\) it is shown that the process \((-t\log X_{ts})_{s>0}\) tends to an extremal process \((\overline{\eta}_s)_{s>0}\) in the sense of convergence of the finite dimensional distributions. Additionally it is also shown that \((z\wedge (-t\log X_{ts}))_{s\geq 0}\) converges weakly to \((z\wedge\overline{\eta}_s)_{s\geq 0}\) in \(\mathcal{D}[0,\infty )\), the space of càdlàg functions equipped with Skorokhod’s \(\mathbf{J}_1\) metric.

MSC:

60G70 Extreme value theory; extremal stochastic processes
60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
60G51 Processes with independent increments; Lévy processes
60J35 Transition functions, generators and resolvents
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bar-Lev, S. K.; Enis, P., Existence of moments and an asymptotic result based on a mixture of exponential distributions, Statist. Probab. Lett., 5, 273-277 (1987), (in English) · Zbl 0622.60030
[2] Bar-Lev, S. K.; Löpker, A.; Stadje, W., On the small-time behavior of subordinators, Bernoulli, 18, 3, 823-835 (2012), (in English) · Zbl 1259.60049
[3] Bertoin, J., Lévy Processes (1998), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, (in English) · Zbl 0938.60005
[4] Dwass, M., Extremal processes, Ann. Math. Stat., 35, 1718-1725 (1964), (in English) · Zbl 0171.38801
[5] Dynkin, E. B., Markov Processes. Vols. I, II (1965), Springer, (in English) · Zbl 0132.37901
[6] Ethier, S. N.; Kurtz, T. G., Markov Processes. Characterization and Convergence (2005), John Wiley & Sons: John Wiley & Sons Hoboken, NJ, (in English) · Zbl 1089.60005
[7] Lamperti, J., On extreme order statistics, Ann. Math. Stat., 35, 1726-1737 (1964), (in English) · Zbl 0132.39502
[8] Resnick, S. I., Extreme Values, Regular Variation, and Point Processes (1987), Springer, (in English) · Zbl 0633.60001
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.