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The characterisation of the normal distribution

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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There are a number of well known theorems on the mutual independence of forms, either linear or quadratic, in normal variables. Some of these theorems can only hold when the system of variables is normal or degenerate and so the possibility of certain forms being independent characterises the normal distribution. The theorems on characterisation have usually been proved by consideration of the necessary properties of the characteristic function. Here we shall be considering the characteristic function of the variables but we shall make more use of cumulant theory than previous authors. To do so we have first to prove that the existence of cumulants of all orders is implied by the independence conditions. The basic theorems we use are those from general cumulant theory and the special theorems of Cramer and of Marcinkiewicz. An advantage of the methods of this paper is that it is possible to show that some of the characterisation theorems require neither of these special theorems. For example, spherical symmetry is a very strong condition and so neither theorem is required whereas both theorems are needed for the most general theorems on the independence of two linear forms. Throughout we take the class of normal distributions to include the degenerate normal, that is, the distribution of a sure variable.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 1 , Issue 3 , August 1960 , pp. 368 - 383
Copyright
Copyright © Australian Mathematical Society 1960

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