Interesting thread! To me the ``reason" for CLT is simply high-dim geometry. Consider unit ball in dim n+1 & slice it at distance x from the origin to get a dim n ball of radius (1-x^2)^{1/2}. The volume of the slice is prop to (1-x^2)^{n/2}~exp(-(1/2)n x^2). Tada the Gaussian!!
https://twitter.com/shoyer/status/1353021554959872001
In other words, for a random point in the ball, the marginal in any direction will converge to a Gaussian (one line calc!). Maybe this doesn't look like your usual CLT. But consider Bernoulli CLT: 1/sqrt(n) sum_i X_i = <X, u>, with X random in {-1,1}^n & u=1/sqrt(n)*(1,..,1).
That is, the Bernoulli CLT is just about the marginal in the direction u of a random point in the hypercube! So instead of geometry of the ball as in first tweet, we need to consider geometry of the cube. But it turns out that all geometries are roughly the same!
Indeed Dvoretzky's theorem tells us that random subspace of any convex body look approximately like a (rescaled) ball!
There is much more to be said about geometric approaches to CLT. The seminal work [Diaconis & Freedman, 1984] is a good starting point: jstor.org/stable/2240961…
There is much more to be said about geometric approaches to CLT. The seminal work [Diaconis & Freedman, 1984] is a good starting point: jstor.org/stable/2240961…
I should add, more directly than Dvoretzky: as long as the norm of the high-dim vector X concentrates, you will be able to repeat the calc of Tweet 1, and hence get a CLT for most directions. This was first observed by Sudakov in 1978 (see www-users.math.umn.edu/~bobko001/pape… for recent dvlp)
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