Convergence of weighted sums of random variables with long-range dependence. (Q1879488)

Sufficient conditions for the convergence in distribution \[ \frac {1}{m^H}\sum _{n=-\infty }^{\infty }f\left(\frac {n}{m}\right)\xi _n \to \int _{\mathbb R} f(u) dB_H(u), \quad m\to \infty , \] are given where \(f\) is a deterministic function, \(\{\xi _n\}_{n\in \mathbb Z}\) is a sequence of random variables with long-range dependence and \(B_H\) denotes the fractional Brownian motion with the Hurst parameter \(H\in (\frac {1}{2}, 1)\). Two applications are given, the first one concerning the asymptotic behaviour of Weierstrass-Mandelbrot process and the second one coming from the ``moving average'' representation of the fractional Brownian motion with another Hurst parameter \(H'\).