Geodesic
Time-Varying Fractionally Integrated Processes with Nonstationary Long Memory
Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 4, pp. 768-792 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: nonstationary long memory, time-varying fractional integration, partial sums, self-similar processes, asymptotically stationary increments. 
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A. Philippe; D. Surgailis; M.-C. Viano. Time-Varying Fractionally Integrated Processes with Nonstationary Long Memory. Teoriâ veroâtnostej i ee primeneniâ, Tome 52 (2007) no. 4, pp. 768-792. http://geodesic.mathdoc.fr/item/TVP_2007_52_4_a7/

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