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Stationary ARCH models: Dependence structure and central limit theorem. (English) Zbl 0986.60030

Let \(\{\xi_k\}\) be i.i.d. nonnegative random variables. The authors deal with the process \(\{X_k\}\) satisfying ARCH(\(\infty\)) equations \(X_k=\rho_k\xi_k\) and \(\rho_k=a+\sum_{j=1}^{\infty} b_j X_{k-j}\) with \(a\geq 0\), \(b_j\geq 0\) \((j=1,2,\dots)\). It is proved that under conditions \(E\xi_0<\infty\) and \(E\xi_0 \sum_{j=1}^{\infty} b_j<1\) there exists a strictly stationary process \(\{X_k\}\) satisfying the ARCH(\(\infty\)) equations such that it is a unique nonanticipative solution. If, in addition, \(E\xi_0^2<\infty\) and \(\sqrt{E\xi_0^2}\sum_{j=1}^{\infty} b_j <\infty\), then the solution \(\{X_k\}\) is also a unique weakly stationary solution of the ARCH(\(\infty\)) equations. Further it is proved that if \(b_j\sim Cj^{-\gamma}\) for some \(\gamma>1\), then the covariance function of \(\{X_k\}\) also decays at the rate \(k^{-\gamma}\). A moving average representation in martingale differences is derived and the central limit theorem is proved. The proofs are based on a Volterra series type expansion of the ARCH(\(\infty\)) process.
Reviewer: J.Anděl (Praha)

MSC:

60G10 Stationary stochastic processes
60F05 Central limit and other weak theorems
91B84 Economic time series analysis
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