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A generalized unimodality

Published online by Cambridge University Press:  14 July 2016

Richard A. Olshen  and
Leonard J. Savage
Richard A. Olshen
Affiliation:
Stanford University
Leonard J. Savage
Affiliation:
Yale University
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Extract

Khintchine (1938) showed that a real random variable Z has a unimodal distribution with mode at 0 iff Z ~ U X (that is, Z is distributed like U X), where U is uniform on [0, 1] and U and X are independent. Isii ((1958), page 173) defines a modified Stieltjes transform of a distribution function F for w complex thus: Apparently unaware of Khintchine's work, he proved (pages 179-180) that F is unimodal with mode at 0 iff there is a distribution function Φ for which . The equivalence of Khintchine's and Isii's results is made vivid by a proof (due to L. A. Shepp) in the next section.

Type
Research Papers
Information
Journal of Applied Probability , Volume 7 , Issue 1 , April 1970 , pp. 21 - 34
Copyright
Copyright © Applied Probability Trust 

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References

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