A new method for obtaining sharp compound Poisson approximation error estimates for sums of locally dependent random variables

Let $X_1,X_2,...,X_n$ be a sequence of independent or locally dependent random variables taking values in $\mathbb{Z}_+$. In this paper, we derive sharp bounds, via a new probabilistic method, for the total variation distance between the distribution of the sum $\sum_{i=1}^nX_i$ and an appropriate Poisson or compound Poisson distribution. These bounds include a factor which depends on the smoothness of the approximating Poisson or compound Poisson distribution. This "smoothness factor" is of order $\mathrm{O}(\sigma ^{-2})$, according to a heuristic argument, where $\sigma ^2$ denotes the variance of the approximating distribution. In this way, we offer sharp error estimates for a large range of values of the parameters. Finally, specific examples concerning appearances of rare runs in sequences of Bernoulli trials are presented by way of illustration.