Vol. 100, No. 2, 1982

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On the maximum of scaled multinomial variables

David Amiel Freedman

Vol. 100 (1982), No. 2, 329–358

Suppose Sn is a sum of n independent, identically distributed, integer-valued random variables. Let pj = P(Sn = j). Take k independent copies of Sn, and let Nj be the number of these sums which are equal to j. In previous papers Persi Diaconis and I studied

maxj(Nj kpj)
maxj(Nj kpj),
where pj is the normal approximation to pj. Likewise, we have studied the histogram as a density estimator. These problems all have a common structure, namely, determining the asymptotic behavior of the maximum of scaled multinomial variables. The object here is to present a general theorem, flexible enough to cover all the cases mentioned above. The form of this theorem may seem a bit arbitrary at first, but it is suggested by the special cases.

Mathematical Subject Classification 2000
Primary: 60F05
Secondary: 62G05, 62G30
Received: 7 January 1981
Published: 1 June 1982
David Amiel Freedman