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

Let $ Y=X_1+\cdots +X_N $ Y=X1+⋯+XN be a sum of a random number of exchangeable random variables, where the random variable N is independent of the $ X_j $ Xj, and the $ X_j $ 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 $ X_j $ 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 $ X_j $ 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 $ X_j $ 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.