Integral representations and convergence of the renewal density

We derive integral representations for the renewal density u associated with a square integrable probability density p on [ 0 , ∞ ) having finite expected value μ. These representations express u in terms of the real and the imaginary part of the Fourier transform of p, considered as a function on t...

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Bibliographic Details
Published inJournal of mathematical analysis and applications Vol. 323; no. 2; pp. 974 - 984
Main Author Stadje, Wolfgang
Format Journal Article
LanguageEnglish
Published San Diego, CA Elsevier Inc 15.11.2006
Elsevier
Subjects
Asymptotic behavior
Distribution theory
Exact sciences and technology
Fourier analysis
Fourier transform
Global analysis, analysis on manifolds
Integral representation
Mathematical analysis
Mathematics
Probability and statistics
Probability theory and stochastic processes
Renewal density
Sciences and techniques of general use
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Fourier transform
Integral representation
Asymptotic behavior
Renewal density
Probability density
Fourier transformation
Mathematical analysis
Integrability
Half plane
Integral representatior
Probability distribution
Convergence
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Summary:We derive integral representations for the renewal density u associated with a square integrable probability density p on [ 0 , ∞ ) having finite expected value μ. These representations express u in terms of the real and the imaginary part of the Fourier transform of p, considered as a function on the lower complex half plane. We use them to give simple global integrability conditions on p under which lim t → ∞ ( u ( t ) − p ( t ) ) = 1 / μ .
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2005.11.003