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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.

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  • $\begingroup$ What's the connection to random graphs? Is there one? $\endgroup$ – Misha Lavrov Jan 15 at 14:45
  • $\begingroup$ I have not stated the detailed relation to random graphs since it's not crucial for the question asked (and i didn't want to include unnecessary information), although some may recognize a connection. However, i came across this issue when facing a random graph problem, where random adjacency matrices arise by applying indicators on the entries above (symmetry and a vanishing diagonal are already provided). $\endgroup$ – Fiff Jan 16 at 9:08
  • $\begingroup$ That's interesting. If you place an edge $ij$ whenever $\langle X_i, X_j\rangle \ge \alpha$ for some threshold $\alpha$, then you get a random geometric graph in the $d$-sphere, for instance. $\endgroup$ – Misha Lavrov Jan 16 at 14:30
  • $\begingroup$ That is exactly the context in which i am studying these type of matrices. I suspected that the total variation distance between the random geometric graph on the $d$-sphere und the unstructured Erdős–Rényi graph might be symmetric around single-edge probability $p=\frac{1}{2}$ and the above would have shown that. But apparently, this approach is just a dead end. $\endgroup$ – Fiff Jan 16 at 21:03
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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 $d$.)

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  • $\begingroup$ Thanks a lot. Embarrassingly, it did not cross my mind to simply check for counter examples. $\endgroup$ – Fiff Jan 16 at 20:54

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