## Persistence

Climate processes, X(i), often show persistence (Chapter 2). This violates the independence assumption made for deriving the GEV and GP distribution models for extremes. For short-memory persistence such as AR(1) processes, typical for climate (Section 2.1), however, this violation does not invalidate GEV or GP estimation when the sampling of the extremes (Section 6.1) is done appropriately. Even for certain types of long-memory persistence, GEV or GP estimation may still be applied. Our exposition follows closely that of Coles (2001b: Chapter 5 therein).

A stationary process {X(i)}™=1 satisfies the condition D(un) if for all 1 < i1 < ■ ■ ■ < ip < j1 < ■ ■ ■ < jq < n with j1 — ip > l > 0, prob {X(ii) < Un,..., X(ip) < Un, X(ji) < Un,..., X(j) < Un} - prob {X(ii) < Un, . . . , X(ip) < Un} x prob{X(ji) < Un,... ,X(jq) < Un} | < a(n,1), (6.20)

where the sequence a(n, 1n) ^ 0 and 1n/n ^ 0 as n ^ to. An independent process X(i) has zero difference, a, in probabilities. The condition D(Un) generalizes this concept. The integer l plays a similar role as the persistence time.

### 6.2.5.2 Extremal index

Consider the stationary process {X(i)}™=1 with persistence and the related process {X*(i)}™=1 without persistence (but identical data distributions). As explained in Section 6.2.1.1, under suitable conditions the distribution of the block maxima of {X*(i)}™=1 approaches a GEV distribution. Denote the distribution function as FGEV,1(x0ut). It may be shown (Leadbetter et al. 1983) that under the same conditions also the distribution of the block maxima of {X(i)}™=1 approaches a GEV distribution, with other parameters ^ and a (but with identical £). Denote this distribution function as FGEV,2(x0ut). It is (Leadbetter et al. 1983)