Saddlepoint expansions for conditional distributions

A saddlepoint expansion is given for conditional probabilities of the form where is an average of n independent bivariate random vectors. A more general version, corresponding to the conditioning on a p – 1-dimensional linear function of a p -dimensional variable is also included. A separate formula is given for the lattice case. The expansion is a generalization of the Lugannani and Rice (1980) formula, which reappears if and are independent. As an example an approximation to the hypergeometric distribution is derived.