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Characterization of a class of second-order density functions

Published online by Cambridge University Press:  14 July 2016

C. B. Mehr
C. B. Mehr*
Affiliation:
Ohio University
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Distributions of some random variables have been characterized by independence of certain functions of these random variables. For example, let X and Y be two independent and identically distributed random variables having the gamma distribution. Laha showed that U = X + Y and V = X | Y are also independent random variables. Lukacs showed that U and V are independently distributed if, and only if, X and Y have the gamma distribution. Ferguson characterized the exponential distribution in terms of the independence of X – Y and min (X, Y). The best-known of these characterizations is that first proved by Kac which states that if random variables X and Y are independent, then X + Y and X – Y are independent if, and only if, X and Y are jointly Gaussian with the same variance. In this paper, Kac's hypotheses have been somewhat modified. In so doing, we obtain a larger class of distributions which we shall call class λ1. A subclass λ0 of λ1 enjoys many nice properties of the Gaussian distribution, in particular, in non-linear filtering.

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Type
Research Papers
Information
Journal of Applied Probability , Volume 4 , Issue 1 , November 1967, pp. 123 - 129
Copyright
Copyright © Applied Probability Trust 

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References

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