A Counterexample in the Approximation Theory of Random Summation

Abstract

Let $X_n, n \in \mathbb{N}$, be independent and identically distributed random variables and $\tau_n$ be random summation indices such that $\tau_n/n \rightarrow \tau > 0$ in probability. It is shown that even if $\tau_n/n$ converges to $\tau$ as quickly as possible (i.e., $\tau_n/n = \tau$) no general approximation orders for suitably normalized random sums $\sum^{\tau_n(\omega)}_{\nu=1} X_\nu(\omega)$ are available. If, however, the limit function $\tau$ is independent of $X_n, n \in \mathbb{N}$, we give a positive approximation result.