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Some effects of trimming on the law of the iterated logarithm

Part of: Limit theorems Nonparametric inference

Published online by Cambridge University Press:  14 July 2016

Harry Kesten  and
Ross Maller
Harry Kesten
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853-7901, USA. Email address: kesten@math.comell.edu
Ross Maller
Affiliation:
Department of Accounting and Finance, University of Western Australia, Nedlands, WA 6097, Australia. Email address: ross.maller@anu.edu.au
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Abstract

We investigate some effects that the ‘light' trimming of a sum Sn = X1 + X2 + · ·· + Xn of independent and identically distributed random variables has on behaviour of iterated logarithm type. Light trimming is defined as removing a constant number of summands from Sn. We consider two versions: (r)Sn, which is obtained by deleting the r largest Xi from Sn, and , which is obtained by deleting the r variables Xi which are largest in absolute value from Sn. We summarise some relevant results from Rogozin (1968), Heyde (1969), and later writers concerning the untrimmed sum, and add some new results concerning trimmed sums. Among other things we show that a general form of the law of the iterated logarithm holds for but not (completely) for .

Keywords

trimmed sumorder statisticsiterated logarithm law

MSC classification

Primary: 60F15: Strong theorems
Secondary: 60F05: Central limit and other weak theorems 62G30: Order statistics; empirical distribution functions
Type
Part 5. Properties of random variables
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 253 - 271
Copyright
Copyright © Applied Probability Trust 2004 

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