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The Factorisation of the rectangular distribution

Published online by Cambridge University Press:  14 July 2016

T. Lewis*
Affiliation:
University College, London

Extract

Questions of the decomposability of distribution functions into real-valued components of bounded variation were discussed by P. Lévy (1964) in relation to the nature of the components, whether non-decreasing (distribution functions in particular) or absolutely continuous (a.c.) or both. Hanson (1965), in a review of Lévy's paper, raised the question of whether or not a rectangular distribution could be decomposed into two a.c. distributions. In fact, D. G. Kendall had conjectured earlier (Kendall (1960)) that no such decomposition is possible. The object of this paper is to state and prove the truth of Kendall's conjecture. “Decomposition” or “factorisation” will be understood throughout the paper to mean decomposition into distributions. Decompositions of the rectangular distribution into one a.c. and one discrete factor are well known (see, e.g., Lukacs (1960) pp. 128–9), and decompositions in which both factors are singular continuous (s.c.) have been discovered by Kendall and by P. M. Lee; it is shown here that no other combinations of factor-type can exist. References to other work on related decomposability properties are given in the papers by Lévy and Kendall cited above.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 

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References

[1] Hanson, D. L. (1965) Mathematical Reviews 30, # 621, 124125. (A review of the paper by Lévy below).Google Scholar
[2] Kendall, D. G. (1960) Review of Characteristic Functions by E. Lukacs. J. R. Statist. Soc. A 123, 484.Google Scholar
[3] Lévy, P. (1964) Remarques sur l'algèbre des produits de composition. C. R. Acad. Sci. Paris 259, 19231927.Google Scholar
[4] Lukacs, E. (1960) Characteristic Functions. Griffin, London.Google Scholar