Page 16 - Textos de Matemática Vol. 47

6 M. I. GOMES
away just before time n. The extremal behaviour of such a stochastic di↵erence sequence was studied in de Haan et al. (1989).
The most striking feature of the ARCH process in (3.1) lies in the fact that although the building RVs, {Wn}, are standard normal, in the max-domain of attraction of the Gumbel law, ⇤(x) := exp( exp( x)), x 2 R, with a light exponential tail (⇠ = 0), Vn has heavy Pareto-like right-tails, i.e. ⇠ > 0.
Let us consider the process in (3.2). If there is a positive real ↵ such that EA↵n=1, EA↵nln+An<1, 0<EBn↵<1,
the process Xn, in (3.2), has an extremal index ✓ given by Z1 Yj !
✓ = ↵ P sup Ai  y 1 y ↵ 1dy. 1 j 1 i=1
Denoting by Z a strict Pareto(↵) RV, with CDF FZ (z) = 1 z ↵, z 1, and noticing that ✓ = E(IA), with ( Yj )
A=Zsup Ai1,
j 1 i=1
de Haan et al. (1989) have simulated the value of the EI of |Vn| for di↵erent val- ues of . In their simulation they have not used the event A, but the equivalent
event
j 1 i=1 j 1 i=1 where E is a standard exponential RV.
Regarding the EI of the ARCH process, one can use the fact that
Vn =d Cn pVn2 , (3.3)
with {Cn} IID RVs, independent of Vn, and such that, for all n 1, P(Cn = 1) = P(Cn = +1) = 1/2. In Table 1 we show a few simulated values of ✓ for Vn and |Vn|.
Table 1. EI simulated values for the absolute values of the ARCH process and the ARCH process.
(Xj )(EXj ) B= lnY+sup lnAi0 = ↵+sup lnAi0 ,

0.1
0.3
0.5
0.7
0.9
✓(Vn)
0.999
0.939
0.835
0.721
0.612
✓(|Vn|)
0.997
0.887
0.727
0.579
0.460