 Sub-Gaussian distribution

A real centered random variable $X$ is sub-Gaussian random variable with parameter $\sigma \ge 0$ if its moment-generating function satisfiesThe parameter is an upper bound on fluctuation size, not necessarily the standard deviation. For $\sigma >0$ and $u>0$, exponential Markov inequality gives $\mathbb{P}(X\ge u)\le \exp (-tu+{\sigma }^2{t}^2/2)$. Choosing $t=u/{\sigma }^2$ and applying the same argument to $-X$ yields the Chernoff boundA 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 $\cosh t\le e^{t^2/2}$. Products of moment-generating functions show that a sum of 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 variables with a common parameter $\sigma$ has tail bound $2e^{-n{u}^2/(2{\sigma }^2)}$. This gives normal-like concentration without assuming a normal distribution. For an uncentered variable, the convention is to apply the condition to $X-\mathbb{E}X$.