# Mass distributions for high dimensional simplex and cross polytope

In this question, it is shown that the radial mass distribution of an $$n$$-cube (i.e. the probability density for the distance from a point selected uniformly from within an $$n$$-cube to the cube's center) is Gaussian for large $$n$$, with mean $$\sqrt{n/3}$$ and variance $$\frac{1}{45\sqrt{3}}$$.

Question: What are the radial mass distributions for an $$n$$-dimensional simplex and cross polytope, as $$n\rightarrow \infty$$?

Some Monte Carlo experimentation suggests that these distributions are not normal. E.g. for $$n=6$$ the histograms of a cube, cross polytope, and simplex (respectively) look like this:

$$\newcommand\bar\overline$$ Let us first find asymptotics of the radial distribution of a random point uniformly distributed on the standard/probability simplex $$S_n:=\{(x_1,\dots,x_n)\in[0,\infty)^n\colon x_1+\dots+x_n=1\}.$$ The distribution of such a random point coincides with that of the random vector $$\frac{(X_1,\dots,X_n)}{X_1+\dots+X_n},$$ where $$X_1,\dots,X_n$$ are iid standard exponential random variables (r.v.'s) (see e.g. Fisher, bottom of p. 55--top of p. 56). So, the distribution of the distance of such a random point from the origin is that of $$R_n:=\frac{\sqrt{X_1^2+\dots+X_n^2}}{X_1+\dots+X_n} \\ =\frac{\sqrt{n\bar{X^2}}}{n\bar X}=\frac{\sqrt{1+2\bar Y+\bar{Y^2}}}{\sqrt n(1+\bar Y)} \\ =\sqrt2\,\frac{\sqrt{1+\bar Y+(\bar{Y^2}-1)/2}}{\sqrt n(1+\bar Y)},$$ where $$\bar X:=\frac1n\,\sum_1^n X_i$$, $$\bar{X^2}:=\frac1n\,\sum_1^n X_i^2$$, $$\bar Y:=\frac1n\,\sum_1^n Y_i$$, $$\bar{Y^2}:=\frac1n\,\sum_1^n Y_i^2$$, $$Y_1:=X_i-1$$, so that the $$Y_i$$'s are iid with $$EY_i=0$$ and $$EY_i^2=Var\,Y_i=1$$.
Now to get the asymptotic distribution of $$R_n$$ (as $$n\to\infty$$) we use the multivariate delta method (see e.g. this paper), which is here basically a linear approximation of $$R_n$$ based on the fact that $$\bar Y$$ and $$\bar{Y^2}-1$$ are $$O_P(1/\sqrt n)$$. So, $$R_n\approx\sqrt2\,\frac{1+\bar Y/2+(\bar{Y^2}-1)/4}{\sqrt n(1+\bar Y)} \\ \approx\frac{\sqrt2}{\sqrt n}\,(1+\bar Y/2+(\bar{Y^2}-1)/4)(1-\bar Y) \\ \approx\frac{\sqrt2}{\sqrt n}\,(1-\bar Y/2+(\bar{Y^2}-1)/4),$$ where $$A_n\approx B_n$$ means $$A_n=(1+O_P(1/\sqrt n))B_n$$.
By the central limit theorem, $$1-\bar Y/2+(\bar{Y^2}-1)/4$$ is approximately normal with mean $$1$$ and variance $$Var(1-Y_1/2+(Y_1^2-1)/4)/n=1/(4n)$$, we conclude that $$R_n$$ is approximately normal with asymptotic mean $$\sqrt2/\sqrt n$$ and asymptotic standard deviation $$(1/\sqrt2)/n$$.
By symmetry, the radial distribution of a random point uniformly distributed on the standard cross polytope coincides with the radial distribution of a random point uniformly distributed on the "sub-probability" simplex $$\tilde S_n:=\{(x_1,\dots,x_n)\in[0,\infty)^n\colon x_1+\dots+x_n\le1\}$$ and thus, by homogeneity, with the distribution of the r.v. $$\rho_n:=R_n B_n,$$ where $$B_n$$ is a r.v. independent of $$R_n$$ and having the beta distribution with parameters $$1$$ and $$n$$, so that $$EB_n=1-O(1/n)$$ and $$Var\,B_n=O(1/n^2)$$. Thus, $$B_n=1+O_P(1/n)$$. We conclude that the asymptotic radial distribution of a random point uniformly distributed on the standard cross polytope is the same as that of a random point uniformly distributed on the standard simplex.