Page 62 - Textos de Matemática Vol. 47

52 M. M. NEVES
Provided that a stationary sequence {Xn}n 1 has limited long-range de- pendence at extreme levels, the maxima of this sequence follow the same dis- tributional limit law as the associated independent sequence, {Yn}n 1, but with other values for the parameters. This result was established by Leadbetter (1983, Theorem 2.5), and fully detailed in the proof and comments there included, see also Beirlant et al. (2004, page 377). It is presented in the following theorem.
Theorem 2.1. If {Xn}n 1 is a stationary sequence with marginal distribution
F, {Yn}n 1 an i.i.d. segquence of r.v.’s with the same distribution F, Mn :=
max(X ,··· ,X ) and Mg:= max(Y ,··· ,Y ), under the D(u ) condition, 1nnn1onn
withun =anx+bn,Pr (Mn bn)/an x ! G1(x)asn !1,fornor- n!1
malizing sequences {an > 0} and {bn }, if and only if, P r {(Mn bn )/an  x} ! G2(x) where G2(x) = G✓1(x), for a constant ✓ such that 0 < ✓  1.
n!1
So, given that G1(·) ⌘ EV⇠(·), the limit law G2(·) ⌘ EV⇠✓(·) is an extreme
value d.f. with location, scale and shape parameters (μ✓, ✓,⇠✓) given by 1 ✓⇠ ⇠
μ✓=μ ⇠ , ✓= ✓ and ⇠✓=⇠,
where (μ, , ⇠) are the location, scale and shape parameters, respectively, of the limit law of the i.i.d sequence.
The extremal index ✓, 0 < ✓  1, is directly related to the clustering of ex- ceedances: ✓ = 1 for i.i.d. sequences and ✓ ! 0 whenever dependence increases. Definitions can be given extending values of ✓ to the case ✓ = 0, but this is a situation out of interest, see Beirlant et al. (2004).
For illustration of the behaviour of a stationary process for some values of ✓ let us consider the following examples:
Example 2.2. Let {Xn}n 1 be the two-dependent sequence defined as Xn = max(Zn+1,Zn), n 1, where {Zi}i 1 are standard exponential i.i.d. r.v.’s. The underlying model for Xn is given by F (x) = (1 exp( x))2, x 0.
Let {Yn}n 1 be a sequence of i.i.d. r.v.’s from the same distribution, i.e., F(y) = (1 exp( y))2, y 0.
Figure 1 shows a size equal to 2 for the clusters of exceedances of high levels by the {Xn} sequence. Actually for that sequence we have ✓ = 1/2. It can also be seen a shrinkage of the largest observations for the 2-dependent sequence, despite of the fact that we have the same model underlying both sequences.
Example 2.3 (Beirlant et al., 2004, Example 10.3). “Max-Autoregressive Pro- cess (ARMAX process)”.