Gamma, Gaussian and Poisson approximations for random sums using size-biased and generalized zero-biased couplings

Let Y=X1+⋯+XN be a sum of a random number of exchangeable random variables, where the random variable N is independent of the Xj, and the Xj are from the generalized multinomial model introduced by Tallis [(1962). The use of a generalized multinomial distribution in the estimation of correlation in discrete data. Journal of the Royal Statistical Society: Series B (Methodological) 24(2), 530–534]. This relaxes the classical assumption that the Xj are independent. Motivated by applications in insurance, we use zero-biased coupling and its generalizations to give explicit error bounds in the approximation of Y by a Gaussian random variable in Wasserstein distance when either the random variables Xj are centred or N has a Poisson distribution. We further establish an explicit bound for the approximation of Y by a gamma distribution in the stop-loss distance for the special case where N is Poisson. Finally, we briefly comment on analogous Poisson approximation results that make use of size-biased couplings. The special case of independent Xj is given special attention throughout. As well as establishing results which extend beyond the independent setting, our bounds are shown to be competitive with known results in the independent case.