Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes
Let \(\{Y_ n\}\), \(n\geq 1\), be a sequence of random variables such that \(Y_ n=A_ nY_{n-1}+B_ n\), \(n\geq 1\), \(Y_ 0\geq 0\), where \((A_ n,B_ n)'s\) are i.i.d. \({\mathbb{R}}^ 2_+\)-valued random pairs. An example of such a sequence is \(\{X^ 2_ n\}\) where \(\{X_ n\}\) is the first order autoregressive conditional heteroscedastic (ARCH) process used in econometric modelling. Now let \(M_ n=\max (Y_ 1,...,Y_ n)\) and for any \(C\subset {\mathbb{R}}_+\) define \(N_ n(C)=\#\{(k/n)\in C:\) \(Y_ k>u_ n\}\) where \(u_ n\) is a suitably chosen high level. In addition to some mild conditions on \((A_ 1,B_ 1)\), suppose there exists an \(\alpha >0\) such that \(E(A_ 1^{\alpha})=1\), and let \(u_ n=xn^{1/\alpha}\). Main results are: (1) \(P(M_ n\leq u_ n)\to \exp (-c\theta x^{-\alpha})\), \(x>0\), as \(n\to \infty\) and (2) \(N_ n\to^{d}N\), as \(n\to \infty\), where N is a compound Poisson process with intensity parameter \(c\theta x^{- \alpha}.\) Expressions for \(\theta\) and the compounding probabilities of N are given. Simulation procedures to compute them are described. They are applied to compute these parameters for the ARCH process. (The authors remark that the value of c is known only for integer values of \(\alpha\).)
scientific article; zbMATH DE number 4113646
scientific article