# 集中不等式

## 马尔可夫不等式

${\displaystyle \mathbb {P} (|X|\geq a)\leq {\frac {\mathbb {E} (|X|)}{a}}.}$

${\displaystyle \mathbb {P} (X\geq a)=\mathbb {P} (\Phi (X)\geq \Phi (a))\leq {\frac {\mathbb {E} (\Phi (X))}{\Phi (a)}}.}$

## 切比雪夫不等式

${\displaystyle \mathbb {P} (|X-\mathbb {E} (X)|\geq a)\leq {\frac {\operatorname {Var} (X)}{a^{2}}},}$

${\displaystyle \operatorname {Var} (X)=\mathbb {E} [(X-\mathbb {E} (X))^{2}].}$

## Hoeffding不等式

Hoeffding不等式适用于有界的随机变量。设有两两独立的一系列随机变量${\displaystyle X_{1},\dots ,X_{n}\!}$。假设对所有的${\displaystyle 1\leq i\leq n}$${\displaystyle X_{i}}$都是几乎有界的变量，即满足：

${\displaystyle \mathbb {P} (X_{i}\in [a_{i},b_{i}])=1.\!}$

${\displaystyle {\overline {X}}={\frac {X_{1}+\cdots +X_{n}}{n}}}$

${\displaystyle \mathbb {P} ({\overline {X}}-\mathbb {E} [{\overline {X}}]\geq t)\leq \exp \left(-{\frac {2t^{2}n^{2}}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\right),\!}$
${\displaystyle \mathbb {P} (|{\overline {X}}-\mathbb {E} [{\overline {X}}]|\geq t)\leq 2\exp \left(-{\frac {2t^{2}n^{2}}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\right),\!}$

## Efron–Stein不等式

Efron–Stein不等式给出了随机变量方差的一个上限估计。设有两两独立的随机变量${\displaystyle X_{1}\dots X_{n}}$${\displaystyle X_{1}'\dots X_{n}'}$，并且对所有的${\displaystyle i}$${\displaystyle X_{i}'}$${\displaystyle X_{i}}$有着相同的分布。那么令${\displaystyle X=(X_{1},\dots ,X_{n}),X^{(i)}=(X_{1},\dots ,X_{i-1},X_{i}',X_{i+1},\dots ,X_{n})}$，则有

${\displaystyle \mathrm {Var} (f(X))\leq {\frac {1}{2}}\sum _{i=1}^{n}E[(f(X)-f(X^{(i)}))^{2}].}$[1]

## 参考来源

1. Stéphane Boucheron, Gabor Lugosi, Olivier Bousquet. Concentration Inequalities (PDF). Université de Paris-Sud, Laboratoire d'Informatique. [2012年9月7日].（英文）
2. ^ Wassily Hoeffding, Probability inequalities for sums of bounded random variables, Journal of the American Statistical Association 58 (301): 13–30, March 1963. (JSTOR)（英文）