Exponential families admitting almost complex structures

Abstract

We discuss exponential families admitting almost complex structures which are parallel relative to an exponential connection (e–connection) or mixture connection (m–connection). The multinomial distribution, negative multinomial distribution and multivariate normal distribution are important examples of the exponential family. We give almost complex structures which are parallel relative to the exponential or mixture connection for these exponential families. Also, we prove spaces of the multinomial distribution and negative multinomial distribution are of constant curvature with respect to the α–connection.