exponential inequalities

A list of exponential inequalities. These can be used to give concentration inequalities and also immediately yield e-values. They can also be used to construct supermartingales and sub-psi processes.

Throughout, let $X$ be a random variable with mean $\mu$. Set $\overline{X}=X-\mu$.

Name	Condition	Bound
Hoeffding	$X\in [a,b]$	$\forall \lambda \in \mathbb{R},\mathbb{E}\exp \{\lambda \text{Xbar}-\frac{{\lambda }^2}{8}(b-a{)}^2\}\le 1$
Bernstein	$|\text{Xbar}|\le c$	$\forall \lambda \in \mathbb{R}\mathbb{E}\exp \{\lambda \text{Xbar}-{\lambda }^2{\sigma }^2\left(\frac{e^{\lambda c}-1-\lambda c}{(\lambda c{)}^2}\right)\}\le 1$
Self-bounding	MGF exists	$\forall \lambda \in \mathbb{R}\mathbb{E}\exp \{\lambda X-\log \mathbb{E}\exp (\lambda X)\}=1$
Bennett	$|X|\le H$ $\mathbb{E}[X^2]\le v^2<\infty$	$\forall \lambda \in (0,\frac{1}{v})\mathbb{E}\exp \{-\lambda \text{Xbar}-\frac{{\mu }^2}{H^2}(e^{\lambda H}-\lambda H-1)\}\le 1$
One-sided Bernstein I	$X\le H$ and $\mathbb{E}[X^2]<\infty$	$\forall |\lambda |\le \frac{1}{2H}\mathbb{E}\exp \{\lambda \text{Xbar}-(e-2){\lambda }^2\mathbb{V}(X)\}\le 1$
One-sided Bernstein II	$X\ge 0$ and $\mathbb{E}[X^2]<\infty$	$\forall \lambda >0\mathbb{E}\exp \{-\lambda \text{Xbar}-{\lambda }^2\mathbb{E}[X^2]/2\}\le 1$
${\psi }_1$ bound	${\Vert X\Vert }_{{\psi }_1}<\infty$	$\forall \Vert \lambda \Vert \le \frac{1}{2{\Vert \overline{X}\Vert }_{{\psi }_1}}\mathbb{E}\exp (\lambda (\overline{X})-4{\lambda }^2{\Vert \overline{X}\Vert }_{{\psi }_1}^2\le 1$.
Delyon	$\mathbb{E}[X^2]<\infty$	$\forall \lambda \in \mathbb{R}\mathbb{E}[\exp \{\lambda \text{Xbar}-\frac{{\lambda }^2}{6}({\text{Xbar}}^2-2\mathbb{E}[{\text{Xbar}}^2])\}]\le 1$
Fan	$X\ge -1$, $\mathbb{E}[X]\le 0$	$\forall \lambda \in [0,1)\mathbb{E}\exp \{\lambda X+X^2(\log (1-\lambda )+\lambda )\}\le 1$
de la Pena	$X-\mu$ is symmetric	$\forall \lambda \in \mathbb{R}\mathbb{E}\exp \{\lambda \overline{X}-\frac{{\lambda }^2}{2}{\overline{X}}^2\}\le 1$
Catoni	$\mathbb{E}[|X{|}^p]<\infty ,p>1$	$\forall \lambda >0\mathbb{E}\exp \{{\varphi }_p(\lambda X)-\frac{{\lambda }^p}{p}\mathbb{E}[|X{|}^p]\}\le 1$ where ${\varphi }_p(x)\le \log (1+x+|x{|}^p/p)$