Time-Varying Fractionally Integrated Processes with Nonstationary Long Memory

Two classes $A(d), B(d)$ of time-varying linear filters are introduced, built from a given sequence $d = (d_t, t \in Z) $ of real numbers, and such that, for constant $d_t \equiv d$, $A(d)=B(d) = (I -L)^{-d}$ is the usual fractional differencing operator. The invertibility relations $B (-d)\,A(d) = A(-d) B(d) = I$ are established. We study the asymptotic behavior of the partial sums of the filtered white noise processes $Y_t = A(d)\,G \varepsilon_t$ and $X_t = B(d)\,G \varepsilon_t$, when $d $ admits limits $\lim_{t \to \pm \infty} d_t = d_\pm \in (0,{\frac{1}{2}}) $, $G$ being a short memory filter. We show that the limit of partial sums is a self-similar Gaussian process, depending on $d_\pm$ and on the sum of the coefficients of $G$ only. The limiting process has either asymptotically stationary increments, or asymptotically vanishing increments and smooth sample paths.