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

Let $Y=X_1+\cdots+X_N$ be a sum of a random number of exchangeable random variables, where the random variable $N$ is independent of the $X_j$, and the $X_j$ are from the generalized multinomial model introduced by Tallis (1962). This relaxes the classical assumption that the $X_j$ are independent. 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 $X_j$ are centred or $N$ has a Poisson distribution. We further establish an explicit bound for the approximation of $Y$ by a gamma distribution in 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 $X_j$ 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.