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On statistical independence and zero correlation in several dimensions

Published online by Cambridge University Press:  09 April 2009

H. O. Lancaster
H. O. Lancaster
Affiliation:
Department of Mathematical Statistics, University of Sydney.
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Bivariate distributions, subject to a condition of φ2 boundedness to be defined later, can be written in a canonical form. Sarmanov [4] used such a form to deduce that two random variables are independent if and only if the maximal correlation of any square summable function, ξ (x1), of the first variable with any square summable function, η(x2), of the second variable is zero. This is equivalent to the condition that the canonical correlations are all zero. The theorem of Sarmanov [4] was proved without any restriction in Lancaster [2] and the proof is now extended to an arbitrary number of dimensions.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 1 , Issue 4 , December 1960 , pp. 492 - 496
Copyright
Copyright © Australian Mathematical Society 1960

References

[1] Lancaster, H. O., The structure of bivariate distributions, Ann. Math. Statist. 29 (1958), 719–736.CrossRefGoogle Scholar
[2] Lancaster, H. O., Zero correlation and independence, Aust. J. Statist. 1 (1959), 53–56.CrossRefGoogle Scholar
[3] Lancaster, H. O., On tests of independence in several dimensions, J. Aust. Math. Soc. 1 (1960), 241–254.CrossRefGoogle Scholar
[4] Sarmanov, O. V., Maximum correlation coefficient (non-symmetrical case), Dokl. Akad. Nauk S.S.S.R. 121 (1958), 52–55.Google Scholar