Subexponential distribution (light-tailed)

A centered random variable $X$ is sub-exponential with parameters $({\nu }^2,b)$ ifwhere $\nu ,b>0$. Unlike a sub-Gaussian random variable, the moment-generating-function bound is needed only near zero. Optimizing exponential Markov inequality over this restricted interval gives a two-sided tail bound of the formwith an absolute constant $c>0$. Small deviations therefore have a quadratic exponent and large deviations a linear exponent. Equivalently, up to constant changes in parameters, $\mathbb{E}{e}^{|X|/K}$ is bounded for some finite scale $K$.

For an exponential random variable $Y$ of mean one, $X=Y-1$ has moment-generating function $e^{-t}/(1-t)$ for $t<1$. Its logarithm is ${\sum }_{k\ge 2}{t}^k/k$, which is at most $t^2$ for $|t|\le 1/2$, proving a sub-exponential bound. The Centered square of a sub-Gaussian random variable is another example. The term here refers to exponential-tail concentration, and differs from the subexponential distribution defined by heavy-tail convolution asymptotics.

If $\mathbb{E}{e}^{tX}\le e^{{\sigma }^2{t}^2/2}$, then $X^2-\mathbb{E}{X}^2$ is a sub-exponential random variable with scale at most a universal constant times ${\sigma }^2$. The Chernoff bound gives $\mathbb{P}(|X|>x)\le 2e^{-x^2/(2{\sigma }^2)}$; integrating this tail bounds $\mathbb{E}|X{|}^{2q}$ by $2(2{\sigma }^2{)}^{q}q!$. Centering preserves a factorial moment bound, so the Bernstein bound for independent sub-exponential variables apply to independent averages of these squares.

A centered random variable $X$ with $\mathbb{E}[X^2]=\nu$ satisfies the Bernstein moment condition with parameter $b$ whenfor every integer $k\ge 2$. Summing its moment-generating series shows that it is sub-exponential with parameters $(2\nu ,2b)$.

If $X$ and $Y$ are sub-Gaussian, without requiring independence between them, then $XY-\mathbb{E}(XY)$ is a sub-exponential random variable; equivalently, $XY$ has a finite exponential absolute moment at some positive scale. The inequality $2|XY|\le X^2+Y^2$ converts exponential-square moment bounds for the factors into an exponential moment bound for the product.

For independent identically distributed pairs $(U_i,V_i)$ whose coordinates are sub-Gaussian with parameter $\sigma /4$, Bernstein's inequality applied to the centered products gives