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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
N. Alon and J. Spencer, “The probabilistic method”, third edition, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons Inc., Hoboken, NJ (2008).
S. Janson, “Poisson approximation for large deviations”, Random Struct. Alg. 1 (1990), 221–229.
S. Janson, T. Łuczak, and A. Ruciński, “An exponential bound for the probability of nonexistence of a specified subgraph in a random graph”, in Random Graphs ’87 (1990), 73–87.
S. Janson, T. Łuczak, and A. Ruciński, “Random graphs”, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York (2000).
S. Janson and L. Warnke, “The lower tail: Poisson approximation revisited”, Random Struct. Alg. Available at arXiv:1406.1248. http://onlinelibrary.wiley.com/advanced/search/results?start=1
O. Riordan and L. Warnke, “The Janson inequalities for general up-sets”, Random Struct. Alg. 46 (2015), 391–395.
L. Warnke, “Upper tails for arithmetic progressions in random subsets”, Israel J. Math., to appear, available at https://www.dpmms.cam.ac.uk/~lw468/aput.pdf.
Acknowledgements
Svante Janson was partly supported by the Knut and Alice Wallenberg Foundation.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-51753-7_12
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-51752-0
Online ISBN: 978-3-319-51753-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)