Vol. 100, No. 2, 1982

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On the maximum difference between the empirical and expected histograms for sums

Persi W. Diaconis and David Amiel Freedman

Vol. 100 (1982), No. 2, 287–327

Suppose Sn is a sum of n independent and identically distributed random variables with E|X13| < . If n is large, Sn is approximately normal. A histogram of k copies of Sn will be close to the normal curve if k is large relative to √n--log n. This paper derives the joint distribution of the location and size of the maximum deviation between this histogram and the probability histogram for Sn. When k is large relative to √n--(log n)3 , the maximum deviation is taken on at a unique location. The location is normally distributed and independent of the size of the maximum deviation, which has a double-exponential distribution. We construct an example, involving Edgeworth-like expansions, to show the behavior changes if E(X12) < but E|X13| = .

Mathematical Subject Classification 2000
Primary: 60F05
Secondary: 62G05, 62G30
Received: 7 January 1981
Published: 1 June 1982
Persi W. Diaconis
Department of Mathematics
Stanford University
Stanford CA 94305-4065
United States
David Amiel Freedman