Multivariate extreme values for dynamical systems

TL;DR

This paper develops the first theory for multivariate extreme value analysis in dynamical systems, focusing on how extremes across the $d$ components interact along orbits. It introduces a dynamically adapted framework with time-dependence conditions and block-structures that yield a limiting distribution $\mu(\mathbb M_n\le \mathbb u_n(\bbtau)) = e^{-\theta(\bbtau)\hat{\Gamma}(\bbtau)}$, and it expresses the extremal dependence via the stable tail dependence function $\Gamma$ and the Pickands function on the simplex. By classifying configurations of maximising sets into common, distinct, linked, and overlapping types, the paper derives diverse dependence patterns, including asymmetries and time-based clustering captured by the extremal index function $\theta(\bbtau)$. The illustrated examples with simple dynamical maps (doubling/tripling) and the two- and three-point configurations demonstrate how spatial overlap and temporal recurrence shape cross-component extremal dependence, with concrete formulas and piecewise linear Pickands functions.

Abstract

We establish a theory for multivariate extreme value analysis of dynamical systems. Namely, we provide conditions adapted to the dynamical setting which enable the study of dependence between extreme values of the components of $\R^d$-valued observables evaluated along the orbits of the systems. We study this cross-sectional dependence, which results from the combination of a spatial and a temporal dependence structures. We give several illustrative applications, where concrete systems and dependence sources are introduced and analysed.

Figures (6)

• Figure 3.1: Graphs of the Pickands dependence functions of the examples in Section \ref{['subsec:non-periodic-case-no-overlap']} on the left, and Section \ref{['ssec:dlp_per']}, on the right.
• Figure 3.2: Graph of the Pickands dependence function of the example in Section \ref{['sssec:triv']} on the left; Graphs of the extremal index function and of the Pickands dependence function of the example in Section \ref{['ssec:overlap_per']}, in the middle and on the right, respectively.
• Figure 3.3: $U_1^{(n)}(\tau_1)$ is the box outlined with a black line, $U_2^{(n)}(\tau_2)$ is the box outlined in green, the first iterate of $U_1^{(n)}(\tau_1)$ corresponds to the black, dashed box, while the second iterate corresponds to the black, dotted box. $A_n^{(q)}$ corresponds to the regions covered by slanted lines. In this case, $h_1<\lambda^2h_2$. Note that for large $n$ the sets $U_i^{(n)}$ are very thin rectangles with rounded tips, but for pictorial simplicity we disregard the semidisks as mentioned in footnote \ref{['footnote1']}.
• Figure 3.4: $U_1^{(n)}(\tau_1)$ is the box outlined with a black line, $U_2^{(n)}(\tau_2)$ is the green box, the first iterate of $U_1^{(n)}(\tau_1)$ corresponds to the black, dashed box, while the second iterate corresponds to the black, dotted box. $A_n^{(q)}$ corresponds to the regions covered by slanted lines. In this case, $h_1>\lambda^2h_2$. The same comment regarding the shape of $U_i^{(n)}$ as in the caption of Figure \ref{['fig:2dcase1']} applies.
• Figure 3.5: On the left is the graph of the extremal index function and on the right is the graph of the Pickands $D$ function, both associated to $\Gamma$ for the cat map example.

Theorems & Definitions (15)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• proof
• Remark 3.1
• Proposition 3.2
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Remark 3.6