Some Conditional Limit Theorems in Exponential Families

Abstract

Consider a sample from a $p$-dimensional exponential family and a random vector whose distribution is the same as that of the sample given a sufficient statistic. General limit theorems for functions of sum-type for such random vectors are obtained; simple sums, linear combinations, $m$-dependent sums, and $U$-statistics are considered. The results are illustrated by some examples.