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The Lower Tail: Poisson Approximation Revisited

  • Svante JansonEmail author
  • Lutz Warnke
Conference paper
Part of the Trends in Mathematics book series (TM, volume 6)

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.

Notes

Acknowledgements

Svante Janson was partly supported by the Knut and Alice Wallenberg Foundation.

References

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden
  2. 2.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK

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