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.
"Some Conditional Limit Theorems in Exponential Families." Ann. Probab. 9 (5) 818 - 830, October, 1981. https://doi.org/10.1214/aop/1176994310