B0845
Title: Approximations for random sums with equally correlated summands
Authors: Fraser Daly - Heriot-Watt University (United Kingdom) [presenting]
Abstract: Let $Y=X_1+\cdots+X_N$ be a sum of a random number of random variables, where the random variable $N$ is independent of the $X_j$. Such random sums arise in many applications, including in the areas of financial risk, hypothesis testing and physics. Classically, the $X_j$ are assumed to be independent, in which case central limit theorems and other distributional approximation results for $Y$ are well known. However, this assumption of independent $X_j$ is unrealistic in many applications. We relax this restriction, instead of assuming that these random variables come from a generalized multinomial model. In this setting, we prove error bounds in Gaussian and Poisson approximations for $Y$ which allow us to investigate the effect of the correlation parameter on the quality of the approximation, while also providing competitive bounds in the special case of independent $X_j$. We also derive error bounds for Gamma approximation in the special case where $N$ has a Poisson distribution. The proofs make use of Stein's method in conjunction with size-biased and zero-biased couplings.
