Timeline for Bounding random process
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jul 14, 2023 at 21:04 | vote | accept | tony | ||
| Jul 13, 2023 at 19:18 | comment | added | Iosif Pinelis | Previous comment continued: If the research you are working on is not of this typical kind, then perhaps indeed this lemma is not useful to you, and you should look for other tools. | |
| Jul 13, 2023 at 19:17 | comment | added | Iosif Pinelis | Previous comment continued: Of course, the lemma is trivially applicable to $E\sup_{t\in T}X_t$ even when $E\sup_{t\in T}X_t<0$ -- in the sense that the lemma does provide an upper bound on $E\sup_{t\in T}X_t$, even if the bound would be trivial when $E\sup_{t\in T}X_t<0$. My answer provides a less trivial application of the lemma. Also, I have to say that, in typical applications of this lemma, $X_t$ would be zero-mean for all $t$, and then $E\sup_{t\in T}X_t\ge\sup_{t\in T}EX_t=0$, so that $E\sup_{t\in T}X_t\ge0$. | |
| Jul 13, 2023 at 19:17 | comment | added | Iosif Pinelis | @chloe : "how does $E\sup_{t\in T}|X_t|$ relate to $E\sup_{t\in T}X_t$?" -- One such relation is $|E\sup_{t\in T}X_t|\le E\sup_{t\in T}|X_t|$. However, the question in your post was not how $E\sup_{t\in T}X_t$ relates to other things. The question was not, either, "to obtain the upper bound on $E\sup_{t\in T}X_t$" (and actually what could the "the" in "the upper bound" possibly refer to?). Rather, your posted question was "on the applicability of this lemma on random process that its $E[\sup X_t]$ is negative". | |
| Jul 13, 2023 at 16:54 | comment | added | tony | for example, for any $t\in T$, $X_t<0$. Then $\sup X_t<0$, thus $E\sup X_t<0$ as well. Since $E\sup|X_t|=E\sup(-X_t)=E(-\inf X_t)$, if we know the upper bound of $E\sup|X_t|\leq\cdots$ (as derived in your answer), then we know the lower bound if $E(\inf X_t)\geq\cdots$. But our goal is to obtain the upper bound of $E(\sup X_t)\leq\cdots$, equivalently $E(-\inf -X_t)\leq\cdots$, and equivalently $E(\inf -X_t)\geq\cdots$. | |
| Jul 13, 2023 at 12:53 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jul 13, 2023 at 12:48 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jul 13, 2023 at 12:30 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jul 13, 2023 at 12:24 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jul 13, 2023 at 12:17 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |