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Fractional integrals of stationary processes and the central limit theorem

Published online by Cambridge University Press:  14 July 2016

M. Rosenblatt
M. Rosenblatt*
Affiliation:
University of California, San Diego
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Abstract

A class of limit theorems involving asymptotic normality is derived for stationary processes whose spectral density has a singular behavior near frequency zero. Generally these processes have ‘long-range dependence’ but are generated from strongly mixing processes by a fractional integral or derivative transformation. Some related remarks are made about random solutions of the Burgers equation.

Keywords

FRACTIONAL INTEGRALSTATIONARY PROCESSCENTRAL LIMIT THEOREMSINGULAR SPECTRUMBURGERS EQUATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 4 , December 1976 , pp. 723 - 732
Copyright
Copyright © Applied Probability Trust 1976 

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References

[1] Davydov, Yu. A. (1970) The invariance principle for stationary processes. Theor. Prob. Appl. 15, 487–498.CrossRefGoogle Scholar
[2] Helson, H. and Sarason, D. (1967) Past and future. Math. Scand. 21, 5–16.CrossRefGoogle Scholar
[3] Mandelbrot, B. B. (1975) Limit theorems on the self-normalized range for weakly and strongly dependent processes. Z. Wahrscheinlichkeitsth. 31, 271–285.CrossRefGoogle Scholar
[4] McLeish, D. L. (1975) Invariance principles for dependent variables. Z. Wahrscheinlichkeitsth. 32, 165–178.Google Scholar
[5] Rosenblatt, M. (1961) Some comments on narrow band-pass filters. Quart. Appl. Math. 18, 387–393.Google Scholar
[6] Rosenblatt, M. (1968) Remarks on the Burgers equation. J. Math. Phys. 9, 1129–1136.Google Scholar
[7] Taqqu, M. (1975) Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitsth. 31, 287–302.Google Scholar
[8] Zygmund, A. (1968) Trigonometric Series. Cambridge University Press.Google Scholar