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Jan 6, 2023 at 13:21 comment added Iosif Pinelis @Cheng-Yu : Thank you for your appreciation.
Jan 6, 2023 at 6:13 vote accept Cheng-Yu
Jan 6, 2023 at 6:12 comment added Cheng-Yu @losif Thank you very much! This answer is very helpful. I'm amazed by your problem solving skills. It's a shame that I don't have enough reputations to upvote your answer.
Jan 6, 2023 at 4:52 history edited Iosif Pinelis CC BY-SA 4.0
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Jan 6, 2023 at 4:28 comment added Iosif Pinelis @Cheng-Yu : I have added details on the $L^2$ thing.
Jan 6, 2023 at 4:27 history edited Iosif Pinelis CC BY-SA 4.0
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Jan 6, 2023 at 2:13 comment added Cheng-Yu And, would you mind elaborating how this step could be done? It seems not that easy to me.
Jan 6, 2023 at 2:03 comment added Cheng-Yu Thanks for your clear explanation. It seems that the critical step to show the second result (converge in $L^2$), is showing that $\mathbb{E}[(g_n(X) - f(X))^2] < \eta \Longrightarrow |Cov[g_n(X), \epsilon_n]| < $ some_function($\eta$). So, if I define the goodness of approximation as $\mathbb{E}[(g_n(X) - f(X))^2] < \eta$, I can get a positive result?
Jan 6, 2023 at 1:04 history edited Iosif Pinelis CC BY-SA 4.0
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Jan 6, 2023 at 0:54 history answered Iosif Pinelis CC BY-SA 4.0