A model for long memory conditional heteroscedasticity

Abstract

or a particular conditionally heteroscedastic nonlinear (ARCH) process for which the conditional variance of the observable sequence $r_t$ is the square of an inhomogeneous linear combination of $r_s, s < t$, we give conditions under which, for integers $l \geq 2, r_t^l$ has long memory autocorrelation and normalized partial sums of $r_t^l$ converge to fractional Brownian motion.