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.
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Acknowledgements
Svante Janson was partly supported by the Knut and Alice Wallenberg Foundation.
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Janson, S., Warnke, L. (2017). The Lower Tail: Poisson Approximation Revisited. In: Díaz, J., Kirousis, L., Ortiz-Gracia, L., Serna, M. (eds) Extended Abstracts Summer 2015. Trends in Mathematics(), vol 6. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-51753-7_12
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DOI: https://doi.org/10.1007/978-3-319-51753-7_12
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