Distribution of $(\langle X_i,X_j\rangle)_{i,j=1}^n$ for $X_k\sim\operatorname{Unif}(S^d)$

Question

For two independent random variables distributed uniformly on a $d$-sphere surface ($X_1,X_2\sim\operatorname{Unif}(S^d)$), it is obvious that $$\langle X_1,X_2\rangle\sim -\langle X_1,X_2\rangle$$ by symmetry of $X_1$ and linearity of the inner product. Can this notion be generalized to the joint distribution of the pairwise scalar combination of $n$ such variables by using symmetries in a clever way? Namely, does $$((1-\delta_{ij})\langle X_i,X_j\rangle)_{i,j=1}^n\sim (-(1-\delta_{ij})\langle X_i,X_j\rangle)_{i,j=1}^n$$ hold? By rotational invarince, one can derive the density of each such entry rather quick, but that does not seem helpful as the dependence/interaction is rather involved.

Answer 1

The thing you want just isn't true: if you have $n=5$ random variables distributed uniformly on $S^1$, for instance, then it's impossible for $\langle X_i, X_j\rangle$ to be negative for all pairs $i\ne j$. However, it's perfectly possible (and happens with probability better than $\frac{1}{4^4}$: the chances that all of $X_1, X_2, \dots, X_5$ land in the same quadrant) that $\langle X_i, X_j\rangle$ is positive for all pairs $i \ne j$.

(The same thing happens in any dimension if you take $n$ large enough as a function of $d$.)