subexponential distribution (light-tailed)

{{Short description|Type of light-tailed probability distribution}}

{{refimprove|date=May 2024}}

In probability theory, one definition of a subexponential distribution is as a probability distribution whose tails decay at an exponential rate, or faster: a real-valued distribution $\backslash cal\; D$ is called subexponential if, for a random variable $X\backslash sim\; \{\backslash cal\; D\}$,

:$\{\backslash Bbb\; P\}(|X|\backslash ge\; x)=O(e^\{-K\; x\})$, for large $x$ and some constant $K>0$.

The subexponential norm, $\backslash |\backslash cdot\backslash |\_\{\backslash psi\_1\}$, of a random variable is defined by

:$\backslash |X\backslash |\_\{\backslash psi\_1\}:=\backslash inf\backslash \; \backslash \{\; K>0\backslash mid\; \{\backslash Bbb\; E\}(e^$

X|/K})\le 2\}, where the infimum is taken to be $+\backslash infty$ if no such $K$ exists. This is an example of a Orlicz norm. An equivalent condition for a distribution $\backslash cal\; D$ to be subexponential is then that {{r|V|at=§2.7}} Subexponentiality can also be expressed in the following equivalent ways:{{r|V|at=§2.7}} $\{\backslash Bbb\; P\}(|X|\backslash ge\; x)\backslash le\; 2\; e^\{-K\; x\},$ for all $x\backslash ge\; 0$ and some constant $K>0$. $\{\backslash Bbb\; E\}(|X|^p)^\{1/p\}\backslash le\; K\; p,$ for all $p\backslash ge\; 1$ and some constant $K>0$. For some constant $K>0$, $\{\backslash Bbb\; E\}(e^\{\backslash lambda\; |X$

) \le e^{K\lambda} for all $0\backslash le\; \backslash lambda\; \backslash le\; 1/K$.

$\{\backslash Bbb\; E\}(X)$ exists and for some constant $K>0$, $\{\backslash Bbb\; E\}(e^\{\backslash lambda\; (X-\{\backslash Bbb\; E\}(X))\})\backslash le\; e^\{K^2\; \backslash lambda^2\}$ for all $-1/K\backslash le\; \backslash lambda\backslash le\; 1/K$.

$\backslash sqrt$

X

is sub-Gaussian.