Moment convergence in conditional limit theorems

Consider a stun Sigma (N)(1) Y(i) of random variables conditioned on a given value of the sum Sigma (N)(1) X(i) of some other variables, where Xi and Yi are dependent but the pairs (Xi, Yi) form an i.i.d. sequence. We consider here the case when each Xi is discrete. We prove, for a triangular array ((X(ni), Y(ni))) of such pairs satisfying certain conditions, both convergence of the distribution of the conditioned sum (after suitable normalization) to a normal distribution, and convergence of its moments. The results are motivated by an application to hashing with linear probing; we give also some other applications to occupancy problems, random forests, and branching processes.