Bennett's inequality

Let X₁, … X_n be independent random variables with finite variance and assume (for simplicity but without loss of generality) they all have zero expected value. Further assume X_i ≤ a almost surely for all i, and define ${\displaystyle S_{n}=\sum _{i=1}^{n}X_{i}}$ and ${\displaystyle \sigma ^{2}=\sum _{i=1}^{n}\operatorname {E} (X_{i}^{2}).}$ Then for any t ≥ 0,

${\displaystyle \Pr \left(S_{n}>t\right)\leq \exp \left(-{\frac {\sigma ^{2}}{a^{2}}}h\left({\frac {at}{\sigma ^{2}}}\right)\right),}$

where h(u) = (1 + u)log(1 + u) – u.[2][3]

For generalizations see Freedman (1975)[4] and Fan, Grama and Liu (2012)[5] for a martingale version of Bennett's inequality and its improvement, respectively.

Hoeffding's inequality only assumes the summands are bounded almost surely, while Bennett's inequality offers some improvement when the variances of the summands are small compared to their almost sure bounds. However Hoeffding's inequality entails sub-Gaussian tails, whereas in general Bennett's inequality has Poissonian tails. In both inequalities, unlike some other inequalities or limit theorems, there is no requirement that the component variables have identical or similar distributions.

• Concentration inequality - a summary of tail-bounds on random variables.