Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T14:24:48.917Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimation of the mean of a stationary time series by sampling

Published online by Cambridge University Press:  14 July 2016

David R. Brillinger
David R. Brillinger*
Affiliation:
University of Exeter and University of California, Berkeley
*
*Institute of Biometry and Community Medicine.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let X(t), – ∞ < t < ∞, be a stationary time series with mean cx. Let 0 < τ1 < τ2 < … < τN ≦ T denote A given sampling times in the interval (0, T]. We determine the asymptotic distribution of the estimate [X(τ1) + … + X(τN)]/N of cx when the sampling times are fixed, satisfying a form of generalised harmonic analysis requirement, and when the sampling times are the times of events of a stationary point process independent of the series X(t). The results obtained may be viewed as non-standard central limit theorems.

Keywords

STATIONARY SERIESPOINT PROCESSESCENTRAL LIMIT THEOREMSCONFIDENCE INTERVALS
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 2 , June 1973 , pp. 419 - 431
Copyright
Copyright © Applied Probability Trust 1973 

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.)

Footnotes

Research partially supported by N.S.F. Grant GP-31411.

References

Beutler, F. J. (1966) Error-free recovery of signals from irregularly spaced samples. SIAM Rev. 8, 328–335.CrossRefGoogle Scholar
Beutler, F. J. (1970) Alias-free randomly timed sampling of stochastic processes. IEEE Trans. Information Theory 16, 147–152.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Brillinger, D. R. (1972) The spectral analysis of stationary interval functions. Proc. Sixth Berkeley Symposium. Vol. I, 483–513. University of California Press. Berkeley.Google Scholar
Brillinger, D. R. and Rosenblatt, M. (1967) Asymptotic theory of estimates of kth order spectra. Spectral Analysis of Time Series, (ed. Harris, B.), 153–188. Wiley, New York.Google Scholar
Daley, D. J. (1971) Weakly stationary point processes and random measures. J. Roy. Statist. Soc. B, 33, 406–428.Google Scholar
Daley, D. J. and Vere-Jones, D. (1972) A summary of the theory of point processes. Proc. Symp. Stochastic Point Processes. (Ed. Lewis, P. A. W.), 299–383. Wiley, New York.Google Scholar
Leonov, V. P. (1960) The use of the characteristic functional and semi-invariants in the ergodic theory of stationary processes. Soviet Math. Dokl. 1, 878–881.Google Scholar
Leonov, V. P. and Shiryaev, A. N. (1959) On a method of calculation of semi-invariants. Theor. Probability Appl. 4, 319–329.Google Scholar
Loynes, R. M. (1970) Aliasing and related questions in stationary processes. Proc. Twelfth Biennial Seminar Can. Math. Congr. (ed. Pyke, R.), 89–98. Can. Math. Congr., Montreal.Google Scholar
Mann, H. B. and Wald, A. (1943) On stochastic limit and order relations. Ann. Math. Statist. 14, 217–226.Google Scholar
Shiryaev, A. N. (1960) Some problems in the spectral theory of higher-order moments, I. Theor. Probability Appl. 5, 265–284.CrossRefGoogle Scholar
Zaremba, S. K. (1968) Some applications of multidimensional integration by parts. Ann. Polon. Math. 21, 85–96.CrossRefGoogle Scholar