Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-29T12:42:43.236Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, T. W. (1955) The integral of a symmetric unimodal function. Proc. Amer. Math. Soc. 6, 170–176.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications Vol. II. Wiley, New York.Google Scholar
Gnedenko, B. V. and Kolmogorov, A. N. (1954) Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, Mass.Google Scholar
Hardy, G. H., Littlewood, J. E. and PóLya, G. (1934) Inequalities. Cambridge University Press, Cambridge.Google Scholar
Hewitt, E. and Stromberg, K. (1965) Real and Abstract Analysis. Springer, New York.Google Scholar
Ibragimov, I. A. (1956) On the composition of unimodal distributions. Theor. Probability Appl. 1, 255–260.Google Scholar
Isii, K. (1958) Note on a characterization of unimodal distributions. Ann. Inst. Statist. Math. IX, 173–184.Google Scholar
Khintchine, A. Y. (1938) On unimodal distributions. (In Russian). Izv. Nauchno-Issled. Inst. Mat. Mekh., Tomsk. Gos. Univ. 2, 1–7.Google Scholar
Phelps, R. R. (1966) Lectures on Choquet's Theorem. Van Nostrand, Princeton.Google Scholar
Saks, S. (1964) Theory of the Integral. 2nd Revised Ed. Dover, New York.Google Scholar
Wintner, A. (1938) Asymptotic Distributions and Infinite Convolutions. Edwards Brothers, Ann Arbor.Google Scholar