[Paper Review] Regularly varying time series in Banach spaces

Thomas Meinguet, Johan Segers|arXiv (Cornell University)|Jan 19, 2010

Point processes and geometric inequalities26 references21 citations

TL;DR

This paper establishes a theory of regularly varying time series in separable Banach spaces, showing that joint regular variation is equivalent to the weak convergence of rescaled time series conditionally on large norms. The key contribution is the introduction of the tail process and spectral process, which characterize extremal dependence across time and space, with applications to extremograms, tail dependence, and extremal indices in functional data analysis under heavy-tailed assumptions.

ABSTRACT

When a spatial process is recorded over time and the observation at a given time instant is viewed as a point in a function space, the result is a time series taking values in a Banach space. To study the spatio-temporal extremal dynamics of such a time series, the latter is assumed to be jointly regularly varying. This assumption is shown to be equivalent to convergence in distribution of the rescaled time series conditionally on the event that at a given moment in time it is far away from the origin. The limit is called the tail process or the spectral process depending on the way of rescaling. These processes provide convenient starting points to study, for instance, joint survival functions, tail dependence coefficients, extremograms, extremal indices, and point processes of extremes. The theory applies to linear processes composed of infinite sums of linearly transformed independent random elements whose common distribution is regularly varying.

Motivation & Objective

• To develop a coherent theory of extremal dependence for time series of functional data where observations are elements of a Banach space.
• To address limitations of classical functional data analysis under heavy-tailed (infinite variance) innovations by replacing second-order moment assumptions with regular variation.
• To characterize extremal dynamics—such as tail dependence, extremograms, and extremal indices—using the tail and spectral processes in infinite-dimensional settings.
• To establish conditions under which linear processes in Banach spaces are regularly varying, particularly when innovations are i.i.d. and regularly varying.
• To unify and extend finite-dimensional extremal dependence concepts (e.g., extremal index, extremogram) to functional time series via the spectral process.

Proposed method

• Define joint regular variation of a stationary time series $(X_t)_{t\in\mathbb{Z}}$ in a separable Banach space $\mathbb{B}$ via weak convergence of $ (X_t/u)_{t\in\mathbb{Z}} $ given $ \|X_0\| > u $ as $ u \to \infty $.
• Introduce the tail process $ (Y_t)_{t\in\mathbb{Z}} $ as the weak limit of the rescaled process, decomposed into radial (determined by the tail index $\alpha$) and angular (spectral process) components.
• Define the spectral process $ (\Theta_t)_{t\in\mathbb{Z}} $ as the weak limit of $ (X_t / \|X_0\|)_{t\in\mathbb{Z}} $ conditionally on $ \|X_0\| > u $, capturing extremal dependence structure.
• Prove that the distributions of the tail and spectral processes are fully determined by their restrictions to $ t \geq 0 $, enabling inference from forward dynamics.
• Establish that regular variation is preserved under bounded linear operators, enabling the analysis of linear processes $ X_t = \sum_{i\in\mathbb{Z}} T_i Z_{t-i} $ with i.i.d. regularly varying innovations $ Z_t $.
• Derive conditions on operator norms $ \|T_i\| $ ensuring almost sure convergence and regular variation of the linear process, using summability conditions and Potter’s theorem.

Experimental results

Research questions

• RQ1 How can extremal dependence in functional time series be characterized when second-order moments do not exist?
• RQ2 What is the connection between joint regular variation of a Banach-space-valued time series and the weak convergence of its rescaled version given a large norm at time zero?
• RQ3 How do the tail and spectral processes capture both temporal and spatial extremal dependence in infinite-dimensional settings?
• RQ4 Under what conditions is a linear process in a Banach space regularly varying when its innovations are i.i.d. and regularly varying?
• RQ5 To what extent can classical extremal dependence tools (e.g., extremograms, extremal indices) be generalized to functional time series using the spectral process?

Key findings

• Joint regular variation of a stationary time series in a separable Banach space is equivalent to the weak convergence of $ (X_t/u)_{t\in\mathbb{Z}} $ given $ \|X_0\| > u $ as $ u \to \infty $, with the limit being the tail process.
• The spectral process $ (\Theta_t)_{t\in\mathbb{Z}} $, defined as the weak limit of $ (X_t / \|X_0\|)_{t\in\mathbb{Z}} $ conditionally on $ \|X_0\| > u $, fully characterizes extremal dependence across time and space.
• The tail process decomposes into a radial component (determined by the tail index $\alpha$ of $ \|X_0\| $) and an angular component (the spectral process), enabling separation of scale and dependence structure.
• For linear processes $ X_t = \sum_{i\in\mathbb{Z}} T_i Z_{t-i} $, regular variation holds if the innovations $ Z_t $ are i.i.d. and regularly varying, and the operator norms satisfy $ \sum_{i\in\mathbb{Z}} \|T_i\|^\alpha < \infty $.
• The spectral process of such a linear process reflects the heuristic that extremal behavior is driven by a single large innovation, with the spectral process concentrating mass on the index of the largest shock.
• The asymptotic behavior of the tail probability $ \operatorname{P}(\|\sum_i X_i\| > x) $ is shown to be asymptotically equivalent to $ \sum_i \operatorname{P}(\|X_i\| > x) $, with convergence rate controlled by the slowly varying function $ V(x) $ and the index $ \alpha $.