Moment convergence in conditional limit theorems

Abstract

Consider a sum ∑1 N Y i of random variables conditioned on a given value of the sum ∑1 N X i of some other variables, where X i and Y i are dependent but the pairs (X i ,Y i ) form an i.i.d. sequence. We consider here the case when each X i 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.