Vol. 100, No. 2, 1982

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On the mode of an empirical histogram for sums

Persi W. Diaconis and David Amiel Freedman

Vol. 100 (1982), No. 2, 373–385

Suppose Sn is a sum of independent, identically distributed random variables, which are integer-valued, with span 1, and have finite fourth moment. If n is large, Sn is approximately normal. An empirical histogram for k copies of Sn will be close to the normal curve provided k √n--log n. Suppose now that √n-(log n)3 k n52. The object of this paper is to determine the asymptotic joint distribution of the location and size of the mode of this histogram. With overwhelming probability, the mode is unique. Its location and size are asymptotically independent. The location is asymptotically normal, while the size is asymptotically double-exponential. For other k’s, the behavior changes. Likewise, the behavior changes if the third moment is finite but the fourth moment infinite.

Mathematical Subject Classification 2000
Primary: 60F05
Secondary: 62G05, 62G30
Received: 8 August 1980
Published: 1 June 1982
Persi W. Diaconis
Department of Mathematics
Stanford University
Stanford CA 94305-4065
United States
David Amiel Freedman