Maximal inequality for a sequence of partial sums of independent random variables

Question

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 $t>0$,$$\mathbb{P}\left(\max_{1\leq i\leq n}|S_i|\geq t \right)\leq 3\max_{1\leq i\leq n}\mathbb{P}\left(|S_i|\geq\frac{t}{3} \right)$$

Answer 1

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.