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Published in:
2017 | OriginalPaper | Chapter
The Lower Tail: Poisson Approximation Revisited
Authors : Svante Janson, Lutz Warnke
Published in: Extended Abstracts Summer 2015
Publisher: Springer International Publishing
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Abstract
The well-known Janson’s inequality gives Poisson-like upper bounds for the lower tail probability \(\mathbb{P}(X\leqslant (1-\varepsilon )\mathbb{E}X)\) when X is the sum of dependent indicator random variables of a special form. In joint work with Svante Janson we showed that, for large deviations, this inequality is optimal whenever X is approximately Poisson, i.e., when the dependencies are weak. For subgraph counts in random graphs, this, e.g., yields new lower tail estimates, extending earlier work (for the special case ɛ = 1) of Janson, Łuczak and Ruciński.