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

TL;DR

This paper develops explicit error bounds for approximating the distribution of random sums with Gaussian and gamma distributions using zero-biased and size-biased couplings, extending classical results to dependent variables.

Contribution

It introduces new bounds for distributional approximations of random sums with dependent variables using advanced coupling techniques, broadening the scope of existing methods.

Findings

Explicit Wasserstein distance bounds for Gaussian approximation.

Gamma approximation bounds in stop-loss distance for Poisson sums.

Results extend to dependent variables and are competitive with classical independent cases.

Abstract

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.