Timeline for Concentration of a modified random walk
Current License: CC BY-SA 4.0
10 events
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| Oct 30, 2018 at 8:42 | history | edited | Pierre PC | CC BY-SA 4.0 |
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| Oct 30, 2018 at 8:41 | comment | added | Pierre PC | (2) Yes, $\alpha$ and $C$ depend on $\varepsilon$ only in the first approach, whereas $K$ and $K'$ depend on $\alpha$ (hence on $\varepsilon$) and $p$. In the second, $\lambda$ depend on $\varepsilon$ and $a$, and $C$ depends essentially on $\alpha$ (because $a\leq\alpha$). | |
| Oct 30, 2018 at 8:31 | comment | added | Pierre PC | I edited my answer to include the comment of @martin-hairer. Concerning yours: (1) I was looking for a function $f$ such that $\mathbb [f_{n+1}|\mathcal F_n]=f_n$ ($f_n=f(X^{(n)}$), at least away from the origin. I had the intuition that such a function would grow exponentially, and then the calculations (see above) made the appearance of $\alpha$ clear. | |
| Oct 30, 2018 at 8:30 | history | edited | Pierre PC | CC BY-SA 4.0 |
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| Oct 30, 2018 at 2:15 | vote | accept | Xi Wu | ||
| Oct 30, 2018 at 2:14 | comment | added | Xi Wu | Thanks and appreciated. I get the solution. Since I am far from being expert I would like to ask some questions for clarifications: (1) How do you come up with the choice of exp(\alpha|X^{(n)}|)? For a while I was considering supermartingale by transforming |X^{(n)}|, but clearly I failed. Is this related to some consideration of moment generating function? (2) (Just to confirm) Regarding constants here, so \alpha, C only depend on \varepsilon, and K only depends on \alpha and p. | |
| Oct 29, 2018 at 21:19 | comment | added | Martin Hairer | By essentially the same argument you'll find that $\mathbb{E} \exp(a|X^{(n)}|)$ remains bounded uniformly in $n$ whenever $a < \alpha$. | |
| Oct 29, 2018 at 16:36 | history | edited | Pierre PC | CC BY-SA 4.0 |
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| Oct 29, 2018 at 16:28 | history | edited | Pierre PC | CC BY-SA 4.0 |
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| Oct 29, 2018 at 16:11 | history | answered | Pierre PC | CC BY-SA 4.0 |