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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!!

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!