Page 14 - Textos de Matemática Vol. 47

4 M. I. GOMES
is Binomial(n, 1 F (un )), and consequently converges towards a Poisson(⌧ ),
as n ! 1, whereas
Xn
SnX = I{Xj>un}
j=1
converges, as n ! 1, towards a compound Poisson. Then P(MnY  un) = P(SnY = 0) ! e ⌧, whereas P(MnX  un) = P(SnX = 0) ! e ✓⌧, as n!1. This means that the intensity ⌧ of the Poisson limiting exceedances, in the IID case, becomes ✓⌧, i.e. the point process limit for the time normalized upcross- ings of high levels is also a Poisson point process but with intensity ✓⌧. Under independence (or even adequate quite weak dependence), this point process converges to a homogeneous Poisson process, as n ! 1, but when there is a slightly stronger local dependence, clusters of exceedances may occur and the limiting process of exceedances may be a compound Poisson process. Indeed, for a large class of weakly dependent processes, an upcrossing is generally followed by a cluster of exceedances and therefore the clusters may be roughly identified by the occurrence of upcrossings. Indeed, Leadbetter and Nandagopalan (1989) proved that the EI can then also be defined as the reciprocal of the ‘mean time of duration of extreme events’, and it is directly related to the exceedances of high levels. We have
✓ = 1 = lim P(X2 un|X1 >un) limiting mean size of clusters n!1
= lim P(X1  un|X2 > un), n!1
where un is a sequence of values such that (2.1) holds. Then,
P(Yn:n  un) = Fn(un) ! e ⌧ and P(Xn:n  un) ! e ✓⌧.
n!1 n!1
Remark 2.2. The limiting distribution of normalized maximum values of both {Xn} and {Yn} is thus of the same type, but there exists a ‘shrinkage’ of maximum values. This leads to ‘clusters of exceedances of high levels’ with a mean size greater than 1. To illustrate these facts, we next provide in Figure 1, sample paths associated with
{2.1} an IID Fr´echet sequence, {Yi}i 1, from F(x) = exp{ x ↵}, x 0, ↵ = 2 (⇠ = 1/2 = 0.5, ✓ = 1),
{2.2} a stationary 2-dependent sequence also with Fr´echet margins and ↵ = 2, defined as
Xi = 2 1/↵ max (Yi, Yi+1) , Yi given in {2.1}, i 1, for which ✓ = 0.5, and