[jesuslop]

Makes sense, if you have a random process with IID variables, the distributions d are the same, so you can view the partial sums process (∑Xi) dynamic as advancing by convolving with d.

In the generating function pic, the distribution is borne as coefficients of polynomials or of formal power series, and convolution is polynomial multiplication. One example of your step is multiplying by (x+1)/2 (Bernoulli trial), and that gives approaches to the gaussian by chunks of normalized binomial coefficients.

Other related limit, described as a functional central limit theorem is given by Donsker's theorem [1], giving the passage from the discrete situation (random walk) to the continous (Brownian motion).

[1] https://en.wikipedia.org/wiki/Donsker%27s_theorem