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Let $X_{1},X_{2},\ldots$ be i.i.d. random variables and let $S_{n}=X_{1}+ \cdots +X_{n}$. Given that $1<a<a'$ and $0<\sigma<\lambda$, how do I show that if $\sup_{1\leq b \leq a'-a}P(|S_{b}|\geq \sigma)\leq \frac{1}{2}$, then $P( \sup_{a\leq b\leq a'} S_{b}\geq \lambda)\leq 2P(S_{a'}\geq \lambda - \sigma)$?

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Please, do not make it look as if you are giving us homework. Show us what you already did and where you got stuck. Also, people will stop answering your questions if you do not accept a single one of them. –  sxd Nov 20 '11 at 4:22
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Just what Dimitri said. Strongly suggested reading: How-to-ask-a-homework-question. –  Did Nov 20 '11 at 9:34

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