A GENERALIZED UNIMODALITY.

The paper introduces a definition -- more exactly, a one parameter family of definitions -- of unimodality for random objects taking values in a finite dimensional vector space. The possibility of a more general range space is briefly mentioned, and some special attention is given to the one dimensional case and its connections with ordinary unimodality. Two characterizations, or alternative definitions, of alpha-unimodality are given. One of these is an extension of Khintchines theorem to alpha-unimodality. The other is related to an inequality discovered by Anderson for a type of unimodality stricter than n-unimodality for an n-dimensional vector space. In more than one dimension, the distribution of an alpha-unimodal vector can be completely singular, but also it can be absolutely continuous. The densities of absolutely continuous alpha-unimodal random vectors are characterized. The notion of alpha-unimodality permits a little to be salvaged from the known disaster that sums of real, independent, unimodal random numbers need not be unimodal. Author