The Lower Tail: Poisson Approximation Revisited

Abstract

The well-known "Janson's inequality" gives Poisson-like upper bounds for the lower tail probability P(X (1 − ε)EX) when X is the sum of dependent indicator random variables of a special form. We show that, for large deviations, this inequality is optimal whenever X is approximately Poisson, i.e., when the dependencies are weak. We also present correlation-based approaches that, in certain symmetric applications, yield related conclusions when X is no longer close to Poisson. As an illustration we, e.g., consider subgraph counts in random graphs, and obtain new lower tail estimates, extending earlier work (for the special case ε = 1) of Janson, Luczak and Ruciński.

Upper Tail Bounds for Stars

“…Since M < n implies M = M , we infer t M r = (M ) r n r p r−2rγ . So, recalling that Ψ t 2 /Λ t 2 /[(np) r−1 µ] by (15) and that µ dn r+1 p r by (20), using D np, p n −1/(1−γ) and γ 1/(16r) we deduce that Ψ D…”

“…This led to the discovery of Janson's inequality [13,14,24], which gives exponential bounds for P(X H (1 − ε)EX H ) that are best possible up to constant factors in the exponent (cf. the recent work of Janson and Warnke [20]).…”

“…Gearing up to bound P(¬T (β, D, εµ)) via Lemma 9, using e = p γ e s and inequality (20) together with the bound s 1/(r−1) s = 1 + log p −γ p −γ (as 1 + x e x ) it follows that…”

“…We now turn to the final inequality of equation (39). By combining (15) and Λ/µ = 1 + (np) r−1 with M = max{t 1/r , t/n r−1 } and t 1−1/r (log n)1 {p<1/n} + µ 1−1/r 1 {p 1/n} , using p n −9 and µ 1−1/r = Ω r (n 1/r (np) r−1 ), see (20), it follows similarly to (31) that…”

On the Lower Tail Variational Problem for Random Graphs

“…The lower tail problem was also recently analysed by Janson and Warnke [18] using different methods (not relating to the variational problem). In the triangle case, for n −1/2 p → 0, they were able to determine the large deviation rate of…”