Spring 2016

The Lower Tail: Poisson Approximation Revisited

Thursday, May 5, 2016 4:00 pm4:45 pm PDT

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Calvin Lab

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. In joint work with Svante Janson we show 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 eps=1) of Janson, Luczak and Rucinski. If time permits, we shall discuss a correlation-inequality based ingredient of the proof, which is closely related to the question of calculating conditional expectations and variances.