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 ≫logn. Suppose now that (logn)3≪ k ≪ n5∕2. 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.
