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On the duality between the behaviour of sums of independent random variables and the sums of their squares

Published online by Cambridge University Press:  24 October 2008

Peter Hall
Peter Hall
Affiliation:
University of Melbourne
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Abstract

Let Xnj, 1 ≤ j ≤ kn, be independent, asymptotically negligible random variables for each n ≥ 1. In certain cases there exists a duality between the behaviour of ΣjXnj and . We extend one of the known forms of this duality, and show that, under mild conditions on the truncated moments of the Xnj, the convergence of to 1 in the mean of order p (p ≥ 1) is equivalent to the convergence of ΣjXnj to the standard normal law, together with the convergence of its 2pth absolute moment to that of a standard normal variable. A similar result holds in the case of convergence to a Poisson law.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 1 , July 1978 , pp. 117 - 121
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

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