Timeline for Random variables with density distributions given by squared Hermite polynomials
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Feb 10, 2023 at 7:51 | vote | accept | Sunia Cortez | ||
| Feb 7, 2023 at 23:00 | comment | added | Iosif Pinelis | Can you show that the convergence takes place, not only for polynomials, but for (say) any bounded continuous functions, so that we can then talk about the convergence in distribution? | |
| Feb 7, 2023 at 22:34 | comment | added | Carlo Beenakker | it's just a definite integral (I do these with Mathematica); there is no limit sign in the last display because that expression is $k$-independent (as one can see by changing variables from $x$ to $x/\sqrt{k}$). | |
| Feb 7, 2023 at 22:33 | comment | added | Iosif Pinelis | Can you detail the "explicit computation"? Also, a limit sign in the last display is apparently missing. | |
| Feb 7, 2023 at 22:25 | comment | added | Carlo Beenakker | the sense in which $p_k(x)$ approximates $P_k(x)$ for large $k$ is as an integral kernel, I have added an example that hopefully explains this. | |
| Feb 7, 2023 at 22:24 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
integral formula
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| Feb 7, 2023 at 22:10 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
deleted 1 character in body
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| Feb 7, 2023 at 21:31 | comment | added | Iosif Pinelis | I do not see your approximation on the WKB approximation page. As for asymptotic expansions on the Hermite polynomials Wikipedia page, I see there clearly different, apparently mutually incompatible expansions, without a statement on the conditions for, or rigorous meaning of, those expansions. In your post too, it is unclear in what sense the approximation holds and what the sign $\propto$ means. | |
| Feb 7, 2023 at 21:02 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
explained how to arrive at this result
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| Feb 7, 2023 at 20:56 | comment | added | Carlo Beenakker | added two ways to arrive at the result, from the WKB approximation of the harmonic oscillator, or from the asymptotic expansion of the Hermite polynomials. | |
| Feb 7, 2023 at 20:54 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
explained how to arrive at this result
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| Feb 7, 2023 at 20:48 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
explained WKB approximation
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| Feb 7, 2023 at 20:37 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
added 5 characters in body
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| Feb 7, 2023 at 20:31 | comment | added | Iosif Pinelis | Do you have a proof of this approximation? | |
| Feb 7, 2023 at 20:20 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |