Regularly varying multivariate time series

TL;DR

This paper develops a theoretical framework for understanding the extremal behavior of multivariate stationary time series through regular variation and tail processes, with applications to finite-order moving averages.

Contribution

It introduces the concept of joint regular variation for multivariate time series and characterizes the tail process, providing explicit descriptions under mixing conditions.

Findings

Tail process decomposes into independent radial and angular parts.

Limit of point processes for extreme events is explicitly described.

Application to multivariate moving averages with random coefficients.

Abstract

A multivariate, stationary time series is said to be jointly regularly varying if all its finite-dimensional distributions are multivariate regularly varying. This property is shown to be equivalent to weak convergence of the conditional distribution of the rescaled series given that, at a fixed time instant, its distance to the origin exceeds a threshold tending to infinity. The limit object, called the tail process, admits a decomposition in independent radial and angular components. Under an appropriate mixing condition, this tail process allows for a concise and explicit description of the limit of a sequence of point processes recording both the times and the positions of the time series when it is far away from the origin. The theory is applied to multivariate moving averages of finite order with random coefficient matrices.