I guess it's a Gaussian, by the central limit theorem.

No, it can't be. This function has compact support, the Gaussian clearly does not.

You get a Gaussian by repeated convolution of the same function (and normalizing the width).

The equivalent question to what I asked is what's the pdf of X1 + X2/2 + X3/4 + ...

where Xi is iid unit uniform.

Do you mean `rect(2x) • 2rect(2x) • 4rect(4x)` .., both so that the limiting function is a dirac delta and the convolutions remain area preserving? Otherwise since the function you're convolving with keeps getting wider wouldn't the result also keep getting wider, so the support is no longer finite?

It might be possible to get a closed-form solution via an approach like [1]. (Out of curiosity, for rect convolved with itself, the intermediaries seem to be knwon as b-splines: https://www.chebfun.org/examples/approx/BSplineConv.html)

[1] https://math.stackexchange.com/questions/1254392/the-maximum...

Yes that is what I meant, thanks!