Sub-Gaussian Distribution

In probability theory, a sub-Gaussian distribution is a probability distribution with strong tail decay. Informally, the tails of a sub-Gaussian distribution are dominated by (i.e. decay at least as fast as) the tails of a Gaussian. This property gives sub-Gaussian distributions their name.

Formally, the probability distribution of a random variable ''$X$'' is called sub-Gaussian if there are positive constant ''C'' such that for every $t \geq 0$,

: $\operatorname(, X, \geq t) \leq 2 \exp$.

Sub-Gaussian properties

Let ''$X$'' be a random variable. The following conditions are equivalent:

# $\operatorname(, X, \geq t) \leq 2 \exp$ for all $t \geq 0$, where $K_1$ is a positive constant;
# \operatorname exp\leq 2, where $K_2$ is a positive constant;
# $\operatorname , X, ^p \leq 2K_3^p \Gamma\left(\frac+1\right)$ for all ''$p \geq 1$'', where $K_3$ is a positive constant.

''Proof''. $(1)\implies(3)$ By the layer cake representation,$\begin \operatorname , X, ^p &= \int_0^\infty \operatorname(, X, ^p \geq t) dt\\ &= \int_0^\infty pt^\operatorname(, X, \geq t) dt\\ &\leq 2\int_0^\infty pt^\exp\left(-\frac\right) dt\\ \end$After a change of variables $u=t^2/K_1^2$, we find that$\begin \operatorname , X, ^p &\leq 2K_1^p \frac\int_0^\infty u^e^ du\\ &= 2K_1^p \frac\Gamma\left(\frac\right)\\ &= 2K_1^p \Gamma\left(\frac+1\right). \end$

$(3)\implies(2)$ Using the Taylor series for $e^x$:$e^x = 1 + \sum_^\infty \frac,$and monotone convergence theorem, we obtain that\begin
\operatorname exp&= 1 + \sum_^\infty \frac\\
&\leq 1 + \sum_^\infty \frac\\
&= 1 + 2 \sum_^\infty \lambda^p K_3^\\
&= 2 \sum_^\infty \lambda^p K_3^-1\\
&= \frac-1 \quad\text\lambda K_3^2 <1,
\endwhich is less than or equal to $2$ for $\lambda \leq \frac$. Take $K_2 \geq 3^K_3$, then\operatorname exp\leq 2.

$(2)\implies(1)$ By Markov's inequality,$\operatorname(, X, \geq t) = \operatorname\left( \exp\left(\frac\right) \geq \exp\left(\frac\right) \right) \leq \frac \leq 2 \exp\left(-\frac\right).$

Definitions

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

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

More equivalent definitions

The following properties are equivalent:
* The distribution of ''$X$'' is sub-Gaussian.
* Laplace transform condition: for some ''B, b > 0'', $\operatorname e^ \leq Be^$ holds for all $\lambda$.
* Moment condition: for some ''K > 0'', $\operatorname , X, ^p \leq K^p p^$ for all $p \geq 1$.
* Moment generating function condition: for some $L>0$, \operatorname exp(\lambda^2 X^2)\leq \exp(L^2\lambda^2) for all $\lambda$ such that $, \lambda, \leq \frac$.
* Union bound condition: for some ''c > 0'', \ \operatorname max\\leq c \sqrt for all ''n > c'', where $X_1, \ldots, X_n$ are i.i.d copies of ''X''.

Examples

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

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

\inf\left\=\frac, and hence ''$X$'' is a sub-Gaussian random variable.

See also

* Platykurtic distribution

Notes

References

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* {{cite news
, first1=O. , last1=Rivasplata
, title=Subgaussian random variables: An expository note
, journal=Unpublished
, year=2012
, url=http://www.stat.cmu.edu/~arinaldo/36788/subgaussians.pdf

* Vershynin, R. (2018)
"High-dimensional probability: An introduction with applications in data science"
(PDF). Volume 47 of ''Cambridge Series in Statistical and Probabilistic Mathematics''. Cambridge University Press, Cambridge.
* Zajkowskim, K. (2020). "On norms in some class of exponential type Orlicz spaces of random variables". ''Positivity. An International Mathematics Journal Devoted to Theory and Applications of Positivity.'' 24(5): 1231--1240. arXiv:1709.02970
doi.org/10.1007/s11117-019-00729-6

Continuous distributions