Sub-Gaussian distribution

Let ${\displaystyle X}$ be a random variable. The following conditions are equivalent:

1. ${\displaystyle \operatorname {P} (|X|\geq t)\leq 2\exp {(-t^{2}/K_{1}^{2})}}$ for all ${\displaystyle t\geq 0}$, where ${\displaystyle K_{1}}$ is a positive constant;
2. ${\displaystyle \operatorname {E} [\exp {(X^{2}/K_{2}^{2})}]\leq 2}$, where ${\displaystyle K_{2}}$ is a positive constant;
3. ${\displaystyle \operatorname {E} |X|^{p}\leq 2K_{3}^{p}\Gamma \left({\frac {p}{2}}+1\right)}$ for all ${\displaystyle p\geq 1}$, where ${\displaystyle K_{3}}$ is a positive constant.

Proof. ${\displaystyle (1)\implies (3)}$ By the layer cake representation,

$${\displaystyle {\begin{aligned}\operatorname {E} |X|^{p}&=\int _{0}^{\infty }\operatorname {P} (|X|^{p}\geq t)dt\\&=\int _{0}^{\infty }pt^{p-1}\operatorname {P} (|X|\geq t)dt\\&\leq 2\int _{0}^{\infty }pt^{p-1}\exp \left(-{\frac {t^{2}}{K_{1}^{2}}}\right)dt\\\end{aligned}}}$$

After a change of variables ${\displaystyle u=t^{2}/K_{1}^{2}}$, we find that

$${\displaystyle {\begin{aligned}\operatorname {E} |X|^{p}&\leq 2K_{1}^{p}{\frac {p}{2}}\int _{0}^{\infty }u^{{\frac {p}{2}}-1}e^{-u}du\\&=2K_{1}^{p}{\frac {p}{2}}\Gamma \left({\frac {p}{2}}\right)\\&=2K_{1}^{p}\Gamma \left({\frac {p}{2}}+1\right).\end{aligned}}}$$

${\displaystyle (3)\implies (2)}$ Using the Taylor series for ${\displaystyle e^{x}}$:

$${\displaystyle e^{x}=1+\sum _{p=1}^{\infty }{\frac {x^{p}}{p!}},}$$

we obtain that

$${\displaystyle {\begin{aligned}\operatorname {E} [\exp {(\lambda X^{2})}]&=1+\sum _{p=1}^{\infty }{\frac {\lambda ^{p}\operatorname {E} {[X^{2p}]}}{p!}}\\&\leq 1+\sum _{p=1}^{\infty }{\frac {2\lambda ^{p}K_{3}^{2p}\Gamma \left(p+1\right)}{p!}}\\&=1+2\sum _{p=1}^{\infty }\lambda ^{p}K_{3}^{2p}\\&=2\sum _{p=0}^{\infty }\lambda ^{p}K_{3}^{2p}-1\\&={\frac {2}{1-\lambda K_{3}^{2}}}-1\quad {\text{for }}\lambda K_{3}^{2}<1,\end{aligned}}}$$

which is less than or equal to ${\displaystyle 2}$ for ${\displaystyle \lambda \leq {\frac {1}{3K_{3}^{2}}}}$. Take ${\displaystyle K_{2}\geq 3^{\frac {1}{2}}K_{3}}$, then

$${\displaystyle \operatorname {E} [\exp {(X^{2}/K_{2}^{2})}]\leq 2.}$$

${\displaystyle (2)\implies (1)}$ By Markov's inequality,

$${\displaystyle \operatorname {P} (|X|\geq t)=\operatorname {P} \left(\exp \left({\frac {X^{2}}{K_{2}^{2}}}\right)\geq \exp \left({\frac {t^{2}}{K_{2}^{2}}}\right)\right)\leq {\frac {\operatorname {E} [\exp {(X^{2}/K_{2}^{2})}]}{\exp \left({\frac {t^{2}}{K_{2}^{2}}}\right)}}\leq 2\exp \left(-{\frac {t^{2}}{K_{2}^{2}}}\right).}$$

A random variable ${\displaystyle X}$ is called a sub-Gaussian random variable if either one of the equivalent conditions above holds.

The sub-Gaussian norm of ${\displaystyle X}$, denoted as ${\displaystyle \Vert X\Vert _{gauss}}$, is defined by

$${\displaystyle \Vert X\Vert _{gauss}=\inf \left\{c>0:\operatorname {E} \left[\exp {\left({\frac {X^{2}}{c^{2}}}\right)}\right]\leq 2\right\},}$$

which is the Orlicz norm of ${\displaystyle X}$ generated by the Orlicz function ${\displaystyle \Phi (u)=e^{u^{2}}-1.}$ By condition ${\displaystyle (2)}$ above, sub-Gaussian random variables can be characterized as those random variables with finite sub-Gaussian norm.

The following properties are equivalent:

• The distribution of ${\displaystyle X}$ is sub-Gaussian.
• Laplace transform condition: for some B, b > 0, ${\displaystyle \operatorname {E} e^{\lambda (X-\operatorname {E} [X])}\leq Be^{\lambda ^{2}b}}$ holds for all ${\displaystyle \lambda }$.
• Moment condition: for some K > 0, ${\displaystyle \operatorname {E} |X|^{p}\leq K^{p}p^{p/2}}$ for all ${\displaystyle p\geq 1}$.
• Moment generating function condition: for some ${\displaystyle L>0}$, ${\displaystyle \operatorname {E} [\exp(\lambda ^{2}X^{2})]\leq \exp(L^{2}\lambda ^{2})}$ for all ${\displaystyle \lambda }$ such that ${\displaystyle |\lambda |\leq {\frac {1}{L}}}$. [1]
• Union bound condition: for some c > 0, ${\displaystyle \ \operatorname {E} [\max\{|X_{1}-\operatorname {E} [X]|,\ldots ,|X_{n}-\operatorname {E} [X]|\}]\leq c{\sqrt {\log n}}}$ for all n > c, where ${\displaystyle X_{1},\ldots ,X_{n}}$ are i.i.d copies of X.

A standard normal random variable ${\displaystyle X\sim N(0,1)}$ is a sub-Gaussian random variable.

Let ${\displaystyle X}$ be a random variable with symmetric Bernoulli distribution. That is, ${\displaystyle X}$ takes values ${\displaystyle -1}$ and ${\displaystyle 1}$ with probabilities ${\displaystyle 1/2}$ each. Since ${\displaystyle X^{2}=1}$, it follows that

$${\displaystyle \Vert X\Vert _{gauss}=\inf \left\{c>0:\operatorname {E} \left[\exp {\left({\frac {X^{2}}{c^{2}}}\right)}\right]\leq 2\right\}=\inf \left\{c>0:\operatorname {E} \left[\exp {\left({\frac {1}{c^{2}}}\right)}\right]\leq 2\right\}={\frac {1}{\sqrt {\ln 2}}},}$$

and hence ${\displaystyle X}$ is a sub-Gaussian random variable.

• Platykurtic distribution

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