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

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