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
maxjNj
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.
