A note on the strong approximation for long memory processes and its application

Abstract

Let {X k ; k≥1} be a long memory process defined by where {ϵ i ;−∞<i<∞} is a doubly infinite sequence of independent and identically distributed random variables and a i ∼ i −α l(i) for some 1/2<α<1 is a sequence of real numbers. Under some mild conditions, a general strong approximation theorem for partial sums of {X k ; k≥1} is derived, where the variances of the innovations may be infinite. As applications, we establish a general law of the iterated logarithm for the long memory processes and investigate the asymptotic properties of the heavy-tailed long memory model and the adjusted range of partial sums for a kind of long memory processes.