Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T03:11:22.091Z Has data issue: false hasContentIssue false
  • English
  • Français

A simple proof of an identity of Good

Published online by Cambridge University Press:  24 October 2008

J. Groeneveld  and
N. G. Van Kampen
J. Groeneveld
Affiliation:
Instituut voor Theoretische Fysica der Rijksuniversiteit, Princetonplein 5, Utrecht 2506, The Netherlands
N. G. Van Kampen
Affiliation:
Instituut voor Theoretische Fysica der Rijksuniversiteit, Princetonplein 5, Utrecht 2506, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Extract

Recently, Good discovered the following curious identity (1). Let θ be a real-valued random variable having cumulants Km ≡ Km(θ) (m = 1, 2, …). Take a set of m statistically independent variables θ(1), θ(2), …, θ(m), each with the same distribution as θ, and form the random variable

where ω is any primitive mth root of unity (e.g. ω = e2πi/m). Then

where 〈.〉 denotes expectation.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 81 , Issue 2 , March 1977 , pp. 183 - 184
Copyright
Copyright © Cambridge Philosophical Society 1977

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

REFERENCE

(1)Good, I. J.A new formula for cumulants. Math. Proc. Cambridge Philos. Soc. 78 (1975), 333.CrossRefGoogle Scholar