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The general theory of canonical correlation and its relation to functional analysis

Published online by Cambridge University Press:  09 April 2009

E. J. Hannan
E. J. Hannan
Affiliation:
School of General Studies, Australian National University, Canberra.
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The classical theory of canonical correlation is concerned with a standard description of the relationship between any linear combination of ρ random variables xs, and any linear combination of q random variables yt insofar as this relation can be described in terms of correlation. Lancaster [1] has extended this theory, for p = q = 1, to include a description of the correlation of any function of a random variable x and any function of a random variable y (both functions having finite variance) for a class of joint distributions of x and y which is very general. It is the purpose of this paper to derive Lancaster's results from general theorems concerning the spectral decomposition of operators on a Hilbert space. These theorems lend themselves easily to the generalisation of the theory to situations where p and q are not finite. In the case of Gaussian, stationary, processes this generalisation is equivalent to the classical spectral theory and corresponds to a canonical reduction of a (finite) sample of data which is basic. The theory also then extends to any number of processes. In the Gaussian case, also, the present discussion-is connected with the results of Gelfand and Yaglom [2] relating to the amount of information in one random process about another.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 2 , Issue 2 , October 1961 , pp. 229 - 242
Copyright
Copyright © Australian Mathematical Society 1961

References

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