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arXiv · 1406.1248

The lower tail: Poisson approximation revisited

Svante Janson·Lutz Warnke
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Abstract

The well-known "Janson's inequality" gives Poisson-like upper bounds for the lower tail probability \Pr(X \le (1-\eps)\E X) when X is the sum of dependent indicator random variables of a special form. We show that, for large deviations, this inequality is optimal whenever X is approximately Poisson, i.e., when the dependencies are weak. We also present correlation-based approaches that, in certain symmetric applications, yield related conclusions when X is no longer close to Poisson. As an illustration we, e.g., consider subgraph counts in random graphs, and obtain new lower tail estimates, extending earlier work (for the special case \eps=1) of Janson, Luczak and Rucinski.

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Svante Janson, Lutz Warnke. 2014-06-05. The lower tail: Poisson approximation revisited. https://doi.org/10.1002/rsa.20590

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