Convergence of weighted sums of random variables with long-range dependence

Suppose that f is a deterministic function, {ξn}n∈Z is a sequence of random variables with long-range dependence and BH is a fractional Brownian motion (fBm) with index H∈(12,1). In this work, we provide sufficient conditions for the convergence1mH∑n=−∞∞fnmξn→∫Rf(u) dBH(u)in distribution, as m→∞. We also consider two examples. In contrast to the case when the ξnʹs are i.i.d. with finite variance, the limit is not fBm if f is the kernel of the Weierstrass–Mandelbrot process. If however, f is the kernel function from the “moving average” representation of a fBm with index H′, then the limit is a fBm with index H+H′−12.