  One of the main differences between the central limit theorem and the Poisson
law of small numbers is that the former possesses the large sample property
(LSP), i.e., the error of normal approximation to the sum of $n$ independent
identically distributed (i.i.d.) random variables is a decreasing function of
$n$. Since 1980's, considerable effort has been devoted to recovering the LSP
for the law of small numbers in discrete random variable approximation. In this
paper, we aim to establish the LSP for the superposition of i.i.d. point
processes.
