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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
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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

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References

REFERENCE

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