2.8 Bernstein’s Inequality
Theorem 2.8.1 Bernstein inequality.
$X_1,…,X_N$ independent, mean-zero, subexpoential. Then $\forall t\ge 0$, \(\mathbb{P}\{\vert\sum_{i=1}^N X_i\vert\ge t\}\le 2\exp(-c\min(\frac{t^2}{\sum_{i=1}^N\Vert X_i\Vert^2_{\psi_1}}, \frac{t}{\max_i\Vert X_i\Vert^2_{\psi_1}}))\)
where $c>0$ is an absolute constant.
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