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Logarithmic Asymptotics for Multidimensional Extremes Under Nonlinear Scalings

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  30 January 2018

K. M. Kosiński  and
M. Mandjes
K. M. Kosiński*
Affiliation:
University of Amsterdam, and Eindhoven University of Technology
M. Mandjes*
Affiliation:
University of Amsterdam, Eindhoven University of Technology, and CWI Amsterdam
*
∗ Postal address: Instytut Matematyczny, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address: kosinski@math.uni.wroc.pl
∗∗ Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, PO Box 94248, 1090 GE Amsterdam, The Netherlands.
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Abstract

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Let W = {W n : n ∈ N} be a sequence of random vectors in R d , d ≥ 1. In this paper we consider the logarithmic asymptotics of the extremes of W , that is, for any vector q > 0 in R d , we find that logP(there exists n ∈ N: W n uq ) as u → ∞. We follow the approach of the restricted large deviation principle introduced in Duffy (2003). That is, we assume that, for every q ≥ 0, and some scalings {a n }, {v n }, (1 / v n )logP(W n /a n ≥ uq ) has a, continuous in q , limit J W (q ). We allow the scalings {a n } and {v n } to be regularly varying with a positive index. This approach is general enough to incorporate sequences W , such that the probability law of W n / a n satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The equations for these asymptotics are in agreement with the literature.

Keywords

Extrema of stochastic processlarge deviation theory

MSC classification

Primary: 60F10: Large deviations
Secondary: 60G70: Extreme value theory; extremal processes

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 68 - 81
Copyright
© Applied Probability Trust 

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