Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

Weakly Stationary Point Processes and Random Measures

D. J. Daley
Journal of the Royal Statistical Society. Series B (Methodological)
Vol. 33, No. 3 (1971), pp. 406-428
Published by: Wiley for the Royal Statistical Society
Stable URL: http://www.jstor.org/stable/2984684
Page Count: 23
  • Read Online (Free)
  • Download ($29.00)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Weakly Stationary Point Processes and Random Measures
Preview not available

Abstract

A weakly stationary point process on the real line is a point process with counting measure N(A) defined of Borel subsets A of the real line such that the first two moments of N(A) are finite for bounded Borel sets and invariant under translation. The expected number of points falling in an interval of length u equals μ u and serves to define the intensity parameter μ of the process. The variance of the number of points falling in an interval of length u defines the variance function V(u) which, with μ, characterizes weakly stationary point processes. $V(\ldot)$ is absolutely continuous with its left- and right-hand derivatives differing on at most a countable dense set, with $V'(0+) = \alpha = \lim_{\theta\rightarrow\infty} \pi G(\lbrack 0,\theta\rbrack)/\theta$ and G(.) a uniquely determined σ-finite spectral measure on [ 0,∞) in terms of which $V(u) = \int_{\lbrack 0,\infty)} (\sin \frac{1}{2} \theta u/\frac{1}{2}\theta)^2 G(d\theta)$. These general results are also true of any weakly stationary signed random measure. To any operation on a point process there corresponds an operation on the spectral measure, and the analogues of superposition, independent random motion of the points, deletion and multiplication of points and of forming cluster processes are exhibited. The paper attempts to expound a rigorous basis for the general theory preceding Bartlett's (1963) development of spectral analysis for point processes.

Page Thumbnails

  • Thumbnail: Page 
406
    406
  • Thumbnail: Page 
407
    407
  • Thumbnail: Page 
408
    408
  • Thumbnail: Page 
409
    409
  • Thumbnail: Page 
410
    410
  • Thumbnail: Page 
411
    411
  • Thumbnail: Page 
412
    412
  • Thumbnail: Page 
413
    413
  • Thumbnail: Page 
414
    414
  • Thumbnail: Page 
415
    415
  • Thumbnail: Page 
416
    416
  • Thumbnail: Page 
417
    417
  • Thumbnail: Page 
418
    418
  • Thumbnail: Page 
419
    419
  • Thumbnail: Page 
420
    420
  • Thumbnail: Page 
421
    421
  • Thumbnail: Page 
422
    422
  • Thumbnail: Page 
423
    423
  • Thumbnail: Page 
424
    424
  • Thumbnail: Page 
425
    425
  • Thumbnail: Page 
426
    426
  • Thumbnail: Page 
427
    427
  • Thumbnail: Page 
428
    428