Logarithmic asymptotics for multidimensional extremes under nonlinear scalings. (English) Zbl 1321.60048

Kosiński, K. M.; Mandjes, M.

J. Appl. Probab. 52, No. 1, 68-81 (2015).

Summary: Let \(\mathbf{W}=\{\mathbf{W}_n: n\in \mathbb{N}\}\) be a sequence of random vectors in \(\mathbb{R}^d\), \(d\geq 1\). In this paper, we consider the logarithmic asymptotics of the extremes of \(\mathbf{W}\), that is, for any vector \(\mathbf{q}>0\) in \(\mathbb{R}^d\), we find \(\log\mathbb{P}(\exists n\in \mathbb{N}: \mathbf{W}_n>u\mathbf{q})\) as \(u\to \infty\). We follow the approach of the restricted large deviation principle introduced in [K. Duffy et al., Ann. Appl. Probab. 13, No. 2, 430–445 (2003; Zbl 1032.60025)]. That is, we assume that for every \(\mathbf{q}\geq 0\) and some scalings \(\{a_n\}\), \(\{v_n\}\), \((1/v_n)\log\mathbb{P}(\mathbf{W}_n/a_n\geq u\mathbf{q})\) has a – continuous in \(q\) – limit \(J_{\mathbf{W}}(\mathbf{q})\). We allow the scalings \(\{a_n\}\) and \(\{v_n\}\) to be regularly varying with a positive index.is approach is general enough to incorporate sequences \(\mathbf{W}\) such that the probability law of \(\mathbf{W}_n/a_n\) satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The equations for these asymptotics are in agreement with the literature.

MSC:

60F10	Large deviations
60G70	Extreme value theory; extremal stochastic processes

Keywords:

large deviation theory; extrema of stochastic processes

Citations:

Zbl 1032.60025

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