Let \((X_i)_{i \in \iota }\) be a family of measurable real random variables on a probability space \((\Omega , \mu )\) that are mutually independent, and let \(s\) be a finite set of indices. Fix a radius \(t_{\max }\), and suppose that for each \(i \in s\) there is a constant \(c_i\) such that \(e^{tX_i}\) is \(\mu \)-integrable and \(\mathbf{E}_\mu [e^{tX_i}] \le e^{c_i t^2}\) for every \(t\) with \(|t| \le t_{\max }\). Then the sum \(\sum _{i \in s} X_i\) satisfies the same two-parameter sub-exponential bound with radius \(t_{\max }\) and constant \(\sum _{i \in s} c_i\): for every \(t\) with \(|t| \le t_{\max }\), the moment \(e^{t \sum _{i \in s} X_i}\) is \(\mu \)-integrable and \(\mathbf{E}_\mu \! \left[e^{t \sum _{i \in s} X_i}\right] \le e^{(\sum _{i \in s} c_i)\, t^2}\).

Let \(X : \Omega \to \mathbb {R}\) be a random variable on a finite measure space \((\Omega , \mu )\), and suppose \(X\) satisfies the Bernstein moment-generating-function condition with parameters \(c\) and \(t_{\max }\): for every \(t\) with \(\left\lvert t\right\rvert \le t_{\max }\) the exponential moment \(e^{tX}\) is \(\mu \)-integrable and \(\mathbb {E}_\mu [e^{tX}] \le e^{c t^2}\). If \(c \gt 0\), then for every \(s\) with \(0 \le s \le 2c\, t_{\max }\) the upper tail obeys

Let \(X : \Omega \to \mathbb {R}\) be a random variable on a probability space carrying a finite measure \(\mu \), and suppose \(X\) satisfies the Bernstein moment generating function condition with parameters \(c\) and \(t_{\max }\): for every \(t\) with \(|t| \le t_{\max }\), the exponential moment \(e^{tX}\) is \(\mu \)-integrable and \(\mathbb {E}_\mu [e^{tX}] \le e^{c t^2}\). If \(c \gt 0\), then for every \(s\) with \(0 \le s \le 2 c\, t_{\max }\),

Let \(X : \Omega \to \mathbb {R}\) be a random variable on a probability space \((\Omega , \mu )\), and let \(c, t_{\max } \ge 0\) be real parameters such that \(X\) satisfies the Bernstein moment generating function condition with parameters \(c\) and \(t_{\max }\): for every \(t\) with \(\left\lvert t\right\rvert \le t_{\max }\) the exponential moment \(e^{tX}\) is \(\mu \)-integrable and \(\mathbb {E}_\mu [e^{tX}] \le e^{c t^2}\). If \(c \gt 0\), then for every \(s\) with \(0 \le s \le 2 c\, t_{\max }\),

Let \((Y_i)_{i}\) be a finite family of measurable real random variables on a probability space \((\Omega , \mu )\), indexed by a finite set of cardinality \(n\), and suppose the \(Y_i\) are mutually independent. Fix real numbers \(\sigma ^2\), \(c \gt 0\), and \(t_{\max }\), and suppose that for each \(i\) the centered square \(Y_i^2 - \sigma ^2\) satisfies the Bernstein MGF condition with parameters \(c\) and \(t_{\max }\). Then for every \(s\) with \(0 \le s \le 2 n c\, t_{\max }\),