Hostname: page-component-7bb8b95d7b-2h6rp Total loading time: 0 Render date: 2024-09-29T19:50:23.103Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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