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Convergence in distribution of lightly trimmed and untrimmed sums are equivalent

Published online by Cambridge University Press:  24 October 2008

Harry Kesten
Harry Kesten
Affiliation:
Department of Mathematics, White Hall, Cornell University, Ithaca NY 14853, U.S.A.
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Abstract

We show that trimming a fixed number of terms from sums of i.i.d. random variables (so-called light trimming) can have only a modest effect on limiting behaviour. More specifically, the trimmed sums, after centralization and normalization, have a limit distribution, if and only if the untrimmed sums have a limit distribution (with the same centralization and normalization constants).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 3 , May 1993 , pp. 615 - 638
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Arov, D. Z. and Bobrov, A. A.. The extreme terms of a sample and their role in the sum of independent variables. Theory Probab. Appl. 5 (1960), 377–396.CrossRefGoogle Scholar
[2]Barnett, V. and Lewis, T.. Outliers in Statistical Data (John Wiley & Sons, 1978).Google Scholar
[3]Csörgö, S., Haeusler, E. and Mason, D. M.. A probabilistic approach to the asymptotic distribution of sums of independent, identically distributed random variables. Adv. Appl. Math. 9 (1988), 259–333.CrossRefGoogle Scholar
[4]Darling, D.. The influence of the maximum term in the addition of independent random variables. Trans. Amer. Math. Soc. 73 (1962), 95–107.CrossRefGoogle Scholar
[5]Esseen, C. G.. On the concentration function of a sum of independent random variables. Z. Wahrsch. Verw. Gebiete 9 (1968), 290–308.CrossRefGoogle Scholar
[6]Feller, W.. On regular variation and local limit theorems. In Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. II·1 (editors LeCam, L. M. and Neyman, J.), (University of California Press, 1967), pp. 373–388.Google Scholar
[7]Feller, W.. An Introduction to Probability Theory, vol. II, 2nd ed. (John Wiley & Sons, 1971).Google Scholar
[8]Gnedenko, B. V. and Kolmogorov, A. N.. Limit Distributions for Sums of Independent Random Variables (Addison-Wesley Publ. Co., 1954).Google Scholar
[9]Hall, P.. On the extreme terms of a sample from the domain of attraction of a stable law. J. London Math. Soc. (2) 18 (1978), 181–191.CrossRefGoogle Scholar
[10]Kesten, H. and Maller, R. A.. Ratios of trimmed sums and order statistics. Ann. Probab. 2 (1992).Google Scholar
[11]Kesten, H. and Maller, R. A.. Infinite limits and infinite limit points of random walks and trimmed sums. Research Report, University of Western Australia (1992).Google Scholar
[12]Kesten, H. and Maller, R. A.. Some results on the effect of trimming the sample sum. Research Report, University of Western Australia (1993).Google Scholar
[13]Lévy, P.. Théorie de léAddition des Variables Aléatoires, 2nd ed. (Gauthier-Villars, 1954).Google Scholar
[14]Maller, R. A.. Asymptotic normality of lightly trimmed sums – a converse. Math. Proc. Cambridge Philos. Soc. 92 (1982), 535–545.CrossRefGoogle Scholar
[15]Mori, T.. The strong law of large numbers when extreme terms are excluded from sums. Z. Wahrsch. Verw. Gebiete 36 (1976), 189–194.CrossRefGoogle Scholar
[16]Mori, T.. Stability for sums of i.i.d. random variables when extreme terms are excluded. Z. Wahrsch. Verw. Gebiete 40 (1977), 159–167.CrossRefGoogle Scholar
[17]Mori, T.. On the limit distributions of lightly trimmed sums. Math. Proc. Cambridge Philos. Soc. 96 (1984), 507–516.CrossRefGoogle Scholar
[18]Renyi, A.. Probability Theory (North-Holland Publ. Co., 1970).Google Scholar