= Subexponential distribution
{disambiguate=light-tailed}
{wiki}
= Sub-exponential random variable
{synonym}
A centered <random variable> $X$ is <sub-exponential random variable>[sub-exponential] with parameters $(\nu^2,b)$ if
$$
\mathbb E e^{tX}\leq e^{\nu^2t^2/2}\qquad(|t|<1/b),
$$
where $\nu,b>0$. Unlike a <sub-Gaussian random variable>[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 form
$$
\mathbb P(|X|\geq u)\leq2\exp[-c\min\{u^2/\nu^2,u/b\}],
$$
with 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 $\sum_{k\geq2}t^k/k$, which is at most $t^2$ for $|t|\leq1/2$, proving a <sub-exponential random variable>[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 (heavy-tailed)> defined by heavy-tail <convolution> asymptotics.
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