Timeline for Random variables with density distributions given by squared Hermite polynomials
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 10, 2023 at 7:51 | vote | accept | Sunia Cortez | ||
| Feb 9, 2023 at 8:20 | comment | added | Sunia Cortez | Dear Professor Beenakker, thank you very much indeed for your detailed answer, and all the comments. I am not so familiar with physics arguments, so I need some time to fully understand the picture. I do not know how to technically "accept the answer"? (About generalizations to other orthogonal polynomials, I was also wondering similarly about discrete examples (Meixner, Krawtchouk...)) | |
| Feb 8, 2023 at 11:52 | comment | added | Carlo Beenakker | the approach I sketched in the answer box generalises to orthogonal polynomials that are eigenfunctions of a second order differential equation; we can then apply the WKB method to find the asymptotics at large eigenvalue; if the differential equation is of the form $-\psi''(x)+V(x)\psi(x)=E\psi(x)$, then the asymptotic probability density is $\propto [E-V(x)]^{-1/2}$. (For the Hermite polynomials of your question one has $E=4k$ and $V(x)=x^2$.) | |
| Feb 8, 2023 at 8:02 | comment | added | Sunia Cortez | Thank you for the indications! Would this be similar for other orthogonal polynomials (Chebyshevn Jacobi, Laguerre...)? | |
| Feb 7, 2023 at 20:20 | answer | added | Carlo Beenakker | timeline score: 0 | |
| S Feb 7, 2023 at 17:08 | review | First questions | |||
| Feb 7, 2023 at 23:35 | |||||
| S Feb 7, 2023 at 17:08 | history | asked | Sunia Cortez | CC BY-SA 4.0 |