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Let $X_n$, $n=1,2,3,...$ be a sequence of independent (not necessarily identically distributed) random variables, let $S_n=\sum_{i=1}^nX_i$. Prove the following maximal inequality: For all $t>0$,$$\mathbb{P}\left(\max_{1\leq i\leq n}|S_i|\geq t \right)\leq 3\max_{1\leq i\leq n}\mathbb{P}\left(|S_i|\geq\frac{t}{3} \right)$$

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closed as off-topic by user21820, Chris Custer, José Carlos Santos, cansomeonehelpmeout, Siong Thye Goh May 6 '18 at 21:17

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This inequality is known as Etemadi's inequality. Here is a proof:

For the disjoint sets $$A_j := \left\{ \max_{1 \leq k <j} |S_k| <3r, |S_j| \geq 3r \right\}, \qquad j=1,\ldots,n$$ we have $$ \left\{ \max_{1 \leq j \leq n} |S_j| \geq 3r \right\} = \bigcup_{j=1}^n A_j.$$ Consequently, by the independence of the random variables, $$\begin{align*} \mathbb{P}\left( \max_{1 \leq j \leq n} |S_j| \geq 3r \right) &\leq \mathbb{P}(|S_n| \geq r) + \sum_{j=1}^{n-1} \mathbb{P}(A_j \cap \{|S_n|<r\}) \\ &\leq \mathbb{P}(|S_n| \geq r) + \sum_{j=1}^{n-1} \mathbb{P}(A_j) \mathbb{P}(|S_n-S_j|>2r) \\ &\leq \mathbb{P}(|S_n| \geq r) + \max_{1 \leq j \leq n} \mathbb{P}(|S_n-S_j|>2r) \\ &\leq 3 \max_{1 \leq j \leq n} \mathbb{P}(|S_j| \geq r). \end{align*}$$

Replacing $3r$ by $t$ finishes the proof.

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  • $\begingroup$ Would the downvoter care to comment? Thanks... $\endgroup$ – saz May 30 '15 at 19:49
  • $\begingroup$ Did you find this proof in Gut's book ? $\endgroup$ – Gabriel Romon May 4 '18 at 16:52
  • $\begingroup$ @GabrielRomon Honestly I've never heard of Gut.. so, no, I didn't find it there. $\endgroup$ – saz May 4 '18 at 16:54

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