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Damn this is genius. Immediately strikes me as true though I can’t see a trivial proof. Don’t you need an extra condition though of the vectors being sparse or something? Is this a well known result?
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If they’re sparse product elements would all be zero with high probability. It might even be true without a sparsity condition via central limit theorem. Ie likely as many product elements being positive as negative, and normally distributed in [-1, 1]