= Sub-Gaussian distribution
{wiki}
= Sub-Gaussian random variable
{synonym}
A real centered <random variable> $X$ is <sub-Gaussian random variable> with parameter $\sigma\geq0$ if its <moment-generating function> satisfies
$$
\mathbb E e^{tX}\leq e^{\sigma^2t^2/2}\qquad(t\in\mathbb R).
$$
The parameter is an upper bound on fluctuation size, not necessarily the standard deviation. For $\sigma>0$ and $u>0$, exponential Markov inequality gives $\mathbb P(X\geq u)\leq\exp(-tu+\sigma^2t^2/2)$. Choosing $t=u/\sigma^2$ and applying the same argument to $-X$ yields the <Chernoff bound>
$$
\mathbb P(|X|\geq u)\leq2e^{-u^2/(2\sigma^2)}.
$$
A centered <normal distribution> of <variance> $\sigma^2$ attains equality in the moment-generating-function condition. A <Rademacher distribution> also satisfies it with $\sigma=1$, since $\cosh t\leq e^{t^2/2}$. Products of <moment-generating functions> show that a sum of <independent random variables>[independent] centered sub-Gaussian <random variables> with parameters $\sigma_i$ is sub-Gaussian with parameter $(\sum_i\sigma_i^2)^{1/2}$. Hence the <sample mean> of $n$ <independent random variables>[independent] variables with a common parameter $\sigma$ has tail bound $2e^{-nu^2/(2\sigma^2)}$. This gives normal-like concentration without assuming a <normal distribution>. For an uncentered variable, the convention is to apply the condition to $X-\mathbb EX$.
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