In the previous chapter, we showed that the sample mean of independent sub-exponential random variables \(X_1, \ldots , X_n\) with the parameter \(\alpha \) has the tail probability \(\displaystyle \mathbb {P} (|\bar {X}_n - \mathbb {E} X| > t) \le = 2e^{- \frac {n}{2} \left ( \frac {t^2}{\alpha ^2} \wedge \frac {t}{\alpha } \right )}, \) where \(x \wedge y = \min (x, y)\) and \(x \vee y = \max (x, y)\) . Therefore, with probability at least \(1- \delta \) , \(\displaystyle \lvert \bar {X}_n - \mathbb {E} X \rvert \le \sqrt {\frac {\alpha ^2}{n} \log \Big (\frac {2}{\delta }}\Big ) \vee \left ( \frac {\alpha }{n} \log \Big (\frac {2}{\delta }\Big ) \right ). \) We can see that the two types of sup-exponential tail probability give us two types of rate: \(O(\alpha / \sqrt {n})\) and \(O(\alpha / n)\) . Although the second term is dominated by the first term, it implies the possibility of giving two types of rates in the concentration inequality. We are going to show a stronger concentration inequality of such type.