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Journal of Applied Probability

Some effects of trimming on the law of the iterated logarithm

Harry Kesten and Ross Maller

Source: J. Appl. Probab. Volume 41A, Issue (2004), 253-271.

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 (r) S̃ n, 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 (r) S̃ n but not (completely) for (r)Sn.

Primary Subjects: 60F15, 60J15
Secondary Subjects: 60F05, 62G30

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.jap/1082552203
Digital Object Identifier: doi:10.1239/jap/1082552203
Mathematical Reviews number (MathSciNet): MR2057578
Zentralblatt MATH identifier: 02109924

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