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.

Epsilon Entropy of Stochastic Processes With Continuous Paths

Edward C. Posner and Eugene R. Rodemich
The Annals of Probability
Vol. 1, No. 4 (Aug., 1973), pp. 674-689
Stable URL: http://www.jstor.org/stable/2959437
Page Count: 16
  • Read Online (Free)
  • Download ($19.00)
  • 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.
Preview not available

Abstract

This paper shows that the epsilon entropy in the sup norm of a wide variety of processes with continuous paths on the unit interval is finite. In fact, the class coincides with the class of processes for which proofs of continuity have been given from a covariance condition. This suggests the conjecture that the epsilon entropy of any process continuous on the unit interval is finite in the sup norm of continuous functions. The epsilon entropy considered in this paper is defined as the minimum Shannon entropy of any partition by sets of diameter at most epsilon of the space of continuous functions on the unit interval, where the probability is the one inherited from the given process. The proof proceeds by constructing partitions and estimating their entropy using probability bounds.

Page Thumbnails

  • Thumbnail: Page 
674
    674
  • Thumbnail: Page 
675
    675
  • Thumbnail: Page 
676
    676
  • Thumbnail: Page 
677
    677
  • Thumbnail: Page 
678
    678
  • Thumbnail: Page 
679
    679
  • Thumbnail: Page 
680
    680
  • Thumbnail: Page 
681
    681
  • Thumbnail: Page 
682
    682
  • Thumbnail: Page 
683
    683
  • Thumbnail: Page 
684
    684
  • Thumbnail: Page 
685
    685
  • Thumbnail: Page 
686
    686
  • Thumbnail: Page 
687
    687
  • Thumbnail: Page 
688
    688
  • Thumbnail: Page 
689
    689