Mathematics – Statistics Theory
Scientific paper
2010-10-08
Bernoulli 2010, Vol. 16, No. 2, 301-330
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.3150/09-BEJ201 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti
Scientific paper
10.3150/09-BEJ201
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.
Boutsikas Michael V.
Vaggelatou Eutichia
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