Moment convergence in conditional limit theorems

Abstract

Consider a sum ∑₁^NY_i of random variables conditioned on a given value of the sum ∑₁^NX_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.