7.8 The Extremal Index

So far our discussions of extreme values are based on the assumption that the data are iid random variables. However, in reality extremal events tend to occur in clusters because of the serial dependence in the data. For instance, we often observe large returns (both positive and negative) of an asset after some news event. In this section we extend the theory and applications of extreme values to cases in which the data form a strictly stationary time series. The basic concept of the extension is extremal index, which allows one to characterize the relationship between the dependence structure of the data and their extremal behavior. Our discussion will be brief. Interested readers are referred to Beirlant et al. (2004, Chapter 10) and Embrechts et al. (1997).

Let x1, x2, … be a strictly stationary sequence of random variables with marginal distribution function F(x). Consider the case of n observations {xi|i = 1, … , n}. As before, let x(n) be the maximum of the data, that is, x(n) = max{xi}. We seek the limiting distribution of (x(n) − βn)/αn for some suitably chosen normalizing constants αn > 0 and βn. If {xi} were iid, Section 7.5 shows that the only possible nondegenerate limits are the extreme value distributions. What is the limiting distribution when {xi} are serially dependent?

To answer this question, we start with a heuristic argument. Suppose that the serial dependence of the stationary series xi decays quickly so that xi and xi+ℓ are essentially ...

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