world:useful-bounds
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| Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
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world:useful-bounds [2022/11/17 17:44] rdouc [Doob's inequalities] |
world:useful-bounds [2022/11/17 21:21] rdouc [Maximal Kolmogorov inequality] |
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| Line 74: | Line 74: | ||
| ==== Maximal Kolmogorov inequality ==== | ==== Maximal Kolmogorov inequality ==== | ||
| + | <WRAP center round tip 80%> | ||
| Let \((M_k)_{k\in\nset}\) be a square integrable \((\mcf_k)_{k\in\nset}\)-martingale. Then, | Let \((M_k)_{k\in\nset}\) be a square integrable \((\mcf_k)_{k\in\nset}\)-martingale. Then, | ||
| \begin{equation} | \begin{equation} | ||
| Line 79: | Line 80: | ||
| \end{equation} | \end{equation} | ||
| - | $\bproof$ | + | </WRAP> |
| + | $\bproof$ | ||
| + | <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, | ||
world/useful-bounds.txt · Last modified: 2022/11/18 11:14 by rdouc
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