Here's what's going on, what will illustrate the central limit theorem (CLT) and what will not: For positive integer n and for i = 1, 2, ..., n, let real valu

Here's what's going on, what will illustrate the central limit theorem (CLT) and what will not: For positive integer n and for i = 1, 2, ..., n, let real valued random variable X(i) be so that P(X(i) = -1 = P(X(i) = 1 ) = 1/2 Assume that {X(1), ..., X(n)} is an independent set of random variables. Easily the expectation E[X(i)] = 0. Let real valued random variable S(n) = X(1) + X(2) + ... + X(n) Easily E[S(n)] = 0. For n = 4, S(n) can take on values -4, -3, ..., 0, 1, ..., 4. The animation in the OP should converge to the density of S(n) where n = 4. By the CLT, as n approaches infinity, the density of ( 1/sqrt(n) ) S(n) will be Gaussian with expectation 0. By the strong law of large numbers, as n approaches infinity, the probability random variable (1/n) S(n) converges to 0 is 1. For increasing n, S(n) is a martingale and is a discrete approximation to Brownian motion. As n approaches infinity, the density of S(n) approaches 0 everywhere on the real line and in its limit is not a density.