The Annals of Probability

Necessary and Sufficient Conditions for Sample Continuity of Random Fourier Series and of Harmonic Infinitely Divisible Processes

M. Talagrand

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Abstract

For very general random Fourier series and infinitely divisible processes on a locally compact Abelian group $G$, a necessary and sufficient condition for sample continuity is given in terms of the convergence of a certain series. This series expresses a control on the covering numbers of a compact neighborhood of $G$ by certain nonrandom sets naturally associated with the Fourier series (resp. the process). In the nonstationary case, we give a necessary Sudakov-type condition for a probability measure in a Banach space to be a Levy measure.

Article information

Source
Ann. Probab., Volume 20, Number 1 (1992), 1-28.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989916

Digital Object Identifier
doi:10.1214/aop/1176989916

Mathematical Reviews number (MathSciNet)
MR1143410

Zentralblatt MATH identifier
0790.60039

JSTOR
links.jstor.org

Subjects
Primary: 60G10: Stationary processes
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60G17: Sample path properties 42A20: Convergence and absolute convergence of Fourier and trigonometric series 43A50: Convergence of Fourier series and of inverse transforms 42A61: Probabilistic methods

Keywords
Stationary processes infinitely divisible covering numbers

Citation

Talagrand, M. Necessary and Sufficient Conditions for Sample Continuity of Random Fourier Series and of Harmonic Infinitely Divisible Processes. Ann. Probab. 20 (1992), no. 1, 1--28. doi:10.1214/aop/1176989916. https://projecteuclid.org/euclid.aop/1176989916


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