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

DOI: 10.2140/pjm.1982.100.287

Abstract

Suppose S_n is a sum of n independent and identically distributed random variables with E|X₁³| < ∞. If n is large, S_n is approximately normal. A histogram of k copies of S_n will be close to the normal curve if k is large relative to 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 S_n. When k is large relative to (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

Milestones

Received: 7 January 1981

Published: 1 June 1982