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Let $Y=X_1+\cdots+X_N$ be a sum of a random number of exchangeable randomvariables, 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 usezero-biased coupling and its generalizations to give explicit error bounds inthe approximation of $Y$ by a Gaussian random variable in Wasserstein distancewhen either the random variables $X_j$ are centred or $N$ has a Poissondistribution. 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 approximationresults that make use of size-biased couplings. The special case of independent$X_j$ is given special attention throughout. As well as establishing resultswhich extend beyond the independent setting, our bounds are shown to becompetitive with known results in the independent case.

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