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Let $\{X_i\}_{i\in\mathbb{N}}$ be i.i.d. symmetric random variables, with $-1\leq X_i\leq 1$, $\mathbb{E}(X_i) =0$, $\mathbb{E}(X_i^2) = 1$. We have that: $$ P\left(\bigcap_{k = 1}^{n}\frac{|\sum_{i = ...

Let $X$ be a nonnegative random variable such that $\mathbf{E} \left[ \exp X \right] < \infty$. For $\theta \leqslant 1$, an appropriate application of Jensen's inequality, yields that \begin{align}...