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world:useful-bounds [2022/11/17 21:19] rdouc [Maximal Kolmogorov inequality] |
world:useful-bounds [2022/11/17 21:21] rdouc [Maximal Kolmogorov inequality] |
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$\bproof$ | $\bproof$ | ||
+ | <note tip> | ||
+ | We provide a complete proof here but actually, we can also apply Doob's inequality below to the non-negative submartingale $M_n^2$ | ||
+ | </note> | ||
Let \(\sigma=\inf\set{k\geq 1}{|M_k| \geq \alpha}\) with the convention that $\inf \emptyset=\infty$. | Let \(\sigma=\inf\set{k\geq 1}{|M_k| \geq \alpha}\) with the convention that $\inf \emptyset=\infty$. | ||
Then, | Then, |