Approximations for random sums with equally correlated summands

Let Y = X1 + … + XN be a sum of a random number of random variables, where the random variable N is independent of the Xj. These random sums arise in the modelling of physical and biological phenomena, as well as in applications in financial risk and hypothesis testing. Classically, the Xj are assumed to be independent, in which case central limit theorems and other distributional approximation results for Y are well known. 

However, this assumption of independent Xj may be unrealistic in some applications. We relax this restriction, instead assuming that these random variables come from a generalized multinomial model. In this setting we investigate the effect of the correlation parameter on error bounds in Gaussian, Gamma and Poisson approximations. Proofs make use of Stein's method in conjunction with size-biased and zero-biased couplings.

Venue: Fulton G20