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A nonlinear model for long-memory conditional heteroscedasticity. (English) Zbl 1350.60032

Summary: We discuss a class of conditionally heteroscedastic time series models satisfying the equation \(r_t = \zeta_t\sigma_t\), where \(\zeta_t\) are standardized i.i.d. r.v., and the conditional standard deviation \(\sigma_t\) is a nonlinear function \(Q\) of inhomogeneous linear combination of past values \(r_s\), \(s < t\), with coefficients \(b_j\) . The existence of stationary solution \(rt\) with finite \(p\)-th moment, \(0< p < \infty\), is obtained under some conditions on \(Q\), \(b_j\) and the \(p\)th moment of \(\zeta_0\). Weak dependence properties of \(r_t\) are studied, including the invariance principle for partial sums of Lipschitz functions of \(r_t\). In the case where \(Q\) is the square root of a quadratic polynomial, we prove that \(r_t\) can exhibit a leverage effect and long memory in the sense that the squared process \(r_t^2\) has long-memory autocorrelation and its normalized partial-sum process converges to a fractional Brownian motion.

MSC:

60F17 Functional limit theorems; invariance principles
60G10 Stationary stochastic processes
60G22 Fractional processes, including fractional Brownian motion
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