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A nonlinear model for long memory conditional heteroscedasticity

We discuss a class of conditionally heteroscedastic time series models satisfying the equation $r_t= ζ_t σ_t$, where $ζ_t$ are standardized i.i.d. r.v.'s and the conditional standard deviation $σ_t$ 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 $r_t$ with finite $p$th moment, $0< p < \infty $ is obtained under some conditions on $Q, b_j$ and $p$th moment of $ζ_0$. Weak dependence properties of $r_t$ are studied, including the invariance principle for partial sums of Lipschitz functions of $r_t$. In the case of quadratic $Q^2$, we prove that $r_t$ can exhibit a leverage effect and long memory, in the sense that the squared process $r^2_t$ has long memory autocorrelation and its normalized partial sums process converges to a fractional Brownian motion.