A New Dependence Condition for Time Series and the Extremal Index of Higher-Order Markov Chains

Abstract

We present a new dependence condition for time series and extend the extremal types theorem. The dependence structure of a stationary sequence is described by a sequence of extremal functions. Under a stability condition for the sequence of extremal functions, we obtain the asymptotic distribution of the sample maximum. As a corollary, we derive a surprisingly simple method for computing the extremal index through a limit of a sequence of extremal coefficients. The results may be used to determine the asymptotic distribution of extreme values from stationary time series based on copulas. We illustrate it with the study of the extremal behaviour of d th-order stationary Markov chains in discrete time with continuous state space. For such sequences we present a way to compute the extremal index from the upper extreme value limit for its joint distribution of d + 1 consecutive variables.