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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.

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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

REFERENCES

(1)Alda, V.A note on the Poisson distribution. Czechoslovak Math. J. 77 (1952), 243–246.CrossRefGoogle Scholar
(2)Bernstein, S.Quelques remarques sur le théorème limite Liapounoff. Dokl. Akad. Nauk SSSR (Comptes rendus) 24 (1939), 3–8.Google Scholar
(3)Brown, B. M.Moments of a stopping rule related to the central limit theorem. Ann. Math. Statist. 40 (1969), 1236–1249.CrossRefGoogle Scholar
(4)Brown, B. M.Characteristic functions, moments and the central limit theorem. Ann. Math. Statist. 41 (1970), 658–664.CrossRefGoogle Scholar
(5)Brown, B. M.A note on convergence of moments. Ann. Math. Statist. 42 (1971), 777–779.CrossRefGoogle Scholar
(6)Chung, K. L.A course in probability theory, 2nd edn (New York, Academic Press, 1974).Google Scholar
(7)Gnedenko, B. V.On the theory of limit theorems for sums of independent random variables. Izvestiya Akad. Nauk SSSR (Ser. Mat.), 181–232 (1939), 643–647.Google Scholar
(8)Gnedenko, B. V. and Kolmogorov, A. N.Limit distributions for sums of independent random variables (Reading, Mass., Addison-Wesley, 1954).Google Scholar
(9)Hall, P. G.On the L p convergence of sums of independent random variables. Proc. Cambridge Philos. Soc. 82 (1977), 439–446.CrossRefGoogle Scholar
(10)Marušin, M. N.Ob uslovijah shodimosti k zakoni Puassona i ego momentam dlja summ nezavisimyh slučai˘nyh veličin. Teor. Verojatnost. i Mat. Statist. 14 (1976), 88–95, 156. (In Russian.)Google Scholar
(11)Raikov, D. A.On a connection between the central limit theorem in the theory of probability and the law of large numbers. Izvestiya Akad. Nauk SSSR (Ser. Mat.) (1938), 323–338.Google Scholar