Timeline for Asymptotics of $\chi_m$-distribution where the degree of freedom $m \to \infty?$
Current License: CC BY-SA 4.0
9 events
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| Sep 4, 2020 at 21:21 | comment | added | Iosif Pinelis | @Learningmath : Yes, the pdf convergence also holds, as can be checked directly. | |
| Sep 4, 2020 at 14:56 | comment | added | Learning math | thanks again! I do wonder if we can also say that the $\sqrt{X_m} = \chi_m$ converges in density to the density of $N_m \sim N(\sqrt{m-1/2}, 1/\sqrt{2}), i.e. lim_{m \to \infty} ||f_{\chi_m} - f_{N_m}||_{L^{\infty}(\mathbb{R})} \to 0, m \to \infty?$ Here $f_Z$ denotes the PDF of the random variable $Z.$ | |
| Aug 20, 2020 at 18:26 | vote | accept | Learning math | ||
| Aug 20, 2020 at 13:08 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Aug 19, 2020 at 21:51 | comment | added | Iosif Pinelis | @Learningmath : The uniform (Kolmogorov) distance between the two cdf's does converge to $0$, actually with rate $O(1/\sqrt m)$; stronger than this nonuniform bounds also hold. See the first linked paper of the two added ones. | |
| Aug 19, 2020 at 21:49 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Aug 19, 2020 at 21:32 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Aug 19, 2020 at 21:23 | comment | added | Learning math | @Ioself: thank you, I'll check out the delta method! Quick question before I do though: will this method imply that the distribution of the two CDF's converge to zero? | |
| Aug 19, 2020 at 21:22 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |