Timeline for What is the limiting marginal distribution of a fixed number of coordinates of a random point drawn uniformly on large-dimensional sphere?

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Oct 26, 2021 at 16:52	answer	added	jlewk		timeline score: 0
Oct 21, 2021 at 12:20	answer	added	Iosif Pinelis		timeline score: 2
Oct 21, 2021 at 12:02	comment	added	dohmatob		(continued) It would then follow from Scheffé's Lemma fr.wikipedia.org/wiki/Lemme_de_Scheff%C3%A9 that the marginal distribution of $(X_1,\ldots,X_k)$ converges to $N(0,I_k)$ weakly.
Oct 21, 2021 at 11:38	comment	added	dohmatob		Indeed, thanks. Reached same conclusion: $(X_1,\ldots,X_k)$ has density $$ p_k(x_1,\ldots,x_k) \propto \begin{cases}(1-\sum_{j=1}^k x_j^2/d)^{(d-k)/2},&\mbox{ if }\sum_{j=1}^k x_j^2 < d,\\ 0,&\mbox{ else.} \end{cases} $$ which converges to $\dfrac{e^{-\sum_{j=1}^k x_j^2/2}}{(2\pi)^{d/2}}$ for any $k=o_d(d)$, since $(1-t/d)^{d+o_d(d)} \to e^{-t}$.
Oct 21, 2021 at 11:33	history	edited	dohmatob	CC BY-SA 4.0	added 2 characters in body
Oct 21, 2021 at 11:33	comment	added	Carlo Beenakker		yes, it is true; the $x_j^2$'s are of order $1/d$, so $(1-\sum_{j=1}^k x_j^2)^{(d-k)/2}\rightarrow \exp\left(-(d/2)\sum_{j=1}^k x_j^2\right)$ for $d\rightarrow\infty$ at fixed $k$; the normalisation constraint $\sum_{j=1}^d x_j^2=1$ introduces correlations between the $x_j$'s that become ineffective if the fraction $k/d\rightarrow 0$.
Oct 21, 2021 at 11:27	history	edited	dohmatob	CC BY-SA 4.0	edited title
Oct 21, 2021 at 11:19	history	edited	dohmatob	CC BY-SA 4.0	edited title
Oct 21, 2021 at 11:12	history	asked	dohmatob	CC BY-SA 4.0