Suppose $x$ is an isotropic random variable in $\mathbb{R}^d$ with $E[\|x\|^2]=d$ and $v$ is some vector. It appears that $\sum_i x_i^2 v_i \approx \sum_i v_i$ when $d \approx \infty$.
What is an easy way of showing it?
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How does the "IE" follow? I think that's not true. – whuber Apr 13 '23 at 17:00
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By "IE", I mean that in $\sum_i x_i^2 v_i$, we can replace random $x_i^2$ with 1's, without changing the result much – Yaroslav Bulatov Apr 13 '23 at 17:29
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But the "well approximated" statement appears to be incorrect. – whuber Apr 13 '23 at 17:47
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1@whuber how about this -- for any linear function $f:\mathbb{R}^d\to\mathbb{R}$ we have $d\approx \infty \implies f(x_1^2,\ldots,x_d^2)\approx f(1, 1, \ldots, 1)$ – Yaroslav Bulatov Apr 13 '23 at 18:13
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Wondering if there's a version of Weak Law of Large Numbers which could apply when there's some constraint on uniformity of $v_i$'s – Yaroslav Bulatov Apr 13 '23 at 19:09