Page 15 - Textos de Matemática Vol. 47

NAZARE´ AND ARCH PROCESSES: EXTREMAL INDEX ESTIMATION 5
{2.3} an autoregressive for maxima (ARMAX) Fr´echet stationary sequence, Xi+1 = max(Xi,Ui), Ui :=( ↵ 1)1/↵ Yi,
with ↵ = 2, {Yi}i 1 given in {2.1} as before, and with = 0.8. Then, the EI is ✓ = 1 1/⇠ = 0.36 (see Alpuim, 1989, and Hall, 1996).
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Figure 1. Sample paths of the processes in {2.1} (top left, ✓ = 1), {2.2} (top right, ✓ = 0.5) and {2.3} (bottom, ✓ = 0.36), all from the same underlying F.
3. The ARCH process
The autoregressive conditional heteroscedastic (ARCH) process introduced
by Engle (1982) is defined by
Vn=Wnq + Vn2 1, n 1, (3.1)
where {Wn} are IID standard normal random variables (RVs), > 0, 0< <1.Then, Xn =Vn2 issuchthat
Xn =AnXn 1 +Bn, n 1, X0 0, (3.2)
An = Wn2, Bn = Wn2, {(An, Bn), n 1} IID R2+-valued random pairs. The stochastic di↵erence equation in (3.2) was introduced by Kesten (1973) and further studied in Vervaat (1979), where several examples are provided, like for instance the stock of material checked at regular time intervals, with An the intrinsic decay or increase of the stock and Bn the quantity added or taken