Abstract
Let X1, X2, …, Xn be a sequence of independent or locally dependent random variables taking values in ℤ+. In this paper, we derive sharp bounds, via a new probabilistic method, for the total variation distance between the distribution of the sum ∑i=1nXi 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 O(σ−2), according to a heuristic argument, where σ2 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.
Citation
Michael V. Boutsikas. Eutichia Vaggelatou. "A new method for obtaining sharp compound Poisson approximation error estimates for sums of locally dependent random variables." Bernoulli 16 (2) 301 - 330, May 2010. https://doi.org/10.3150/09-BEJ201
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