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Suppose $S_n, n\geq 0$ is a martingale on $\mathbb{R}$ such that $S_0=0$ and $|S_{n+1}-S_{n}|\in [\frac{1}{2}, 1]$. Prove that there exists $c,C>0$ s.t. $$ \frac{c}{\sqrt{n}} \leq P( S_1\geq 0,\dots S_n\geq 0) \leq \frac{C}{\sqrt{n}}.$$

I actually have no idea in how to approach this. In the case of simple random walk, we can use some reflection principle, but what about here? I don't have any clues.

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  • $\begingroup$ Reflection principle should work here, since you consider a symmetric random walk. Using a backward scheme, remark that you can cross the origin at step $n+1$ only if $S_n<1$. $\endgroup$ – Tom-Tom Sep 16 '15 at 9:37
  • $\begingroup$ Maybe I wrongly use the word "here". I mean how to deal with the generalization rather than how to deal with the case of simple random walk. $\endgroup$ – Brian Ding Sep 16 '15 at 21:35

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