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An inequality for characteristic functions

Published online by Cambridge University Press:  17 April 2009

C.R. Heathcote  and
J.W. Pitman
C.R. Heathcote
Affiliation:
Department of Statistics, School of General Studies, Australian National UniversityCanberra, ACT.
J.W. Pitman
Affiliation:
Department of Statistics, School of General Studies, Australian National UniversityCanberra, ACT.
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Abstract

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The paper is concerned with an extension of the inequality 1 - u(2nt) ≤ 4n[1-u(t)] for u(t) the real part of a characteristic function. The main result is that the inequality in fact holds for all positive integer n and not only powers of 2. Certain consequences are deduced and a brief discussion is given of the circumstances under which equality holds.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 6 , Issue 1 , February 1972 , pp. 1 - 9
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Cramér, Harald, Random variables and probability distributions, 3rd ed. (Cambridge University Press, Cambridge, 1970).Google Scholar
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[3]Kuczma, Marek, Functional equations in a single variable (Polish Scientific Publishers, Warsaw, 1968).Google Scholar
[4]Lukacs, Eugene, Characteristic functions, 2nd ed. (Griffin, London, 1970).Google Scholar