There is a rich field of research on statistical laws (such as the CLT) for deterministic systems, which I think might interest various commenters. Here one starts with a randomly chosen point x in some state space X, some dynamics T on X, and a real (or complex) valued observable function g :X → R and considers the statistics of Y_n = g(T^n(x)) for n > 0 (i.e. we start with X, apply T some number of times and then take a ‘measurement’ of the system via g). In some interesting circumstances these (completely deterministic) systems satisfy a CLT. Specifically, if we set S_n to be the sum of Y_0,… ,Y_{n-1} then for e.g. expanding or hyperbolic T, one can show that S_n is asymptotically a standard normal RV after appropriate scaling/translation to normalise. The key technical hypothesis is that the an invariant measure mu for T exists such that the mu-correlations between Y_0 and Y_k decays summably fast with k.
This also provides an example of a pathological case for the CLT. Specifically, if g is of the form g(x) = h(T(x))-h(x) (a coboundary) with h uniformly bounded then by telescoping the terms in S_n we see that S_n is uniformly bound over n, so when one divides by sqrt(n) the only limit is 0. Thus the limiting distribution is a dirac delta at 0.
There is a rich field of research on statistical laws (such as the CLT) for deterministic systems, which I think might interest various commenters. Here one starts with a randomly chosen point x in some state space X, some dynamics T on X, and a real (or complex) valued observable function g :X → R and considers the statistics of Y_n = g(T^n(x)) for n > 0 (i.e. we start with X, apply T some number of times and then take a ‘measurement’ of the system via g). In some interesting circumstances these (completely deterministic) systems satisfy a CLT. Specifically, if we set S_n to be the sum of Y_0,… ,Y_{n-1} then for e.g. expanding or hyperbolic T, one can show that S_n is asymptotically a standard normal RV after appropriate scaling/translation to normalise. The key technical hypothesis is that the an invariant measure mu for T exists such that the mu-correlations between Y_0 and Y_k decays summably fast with k.
This also provides an example of a pathological case for the CLT. Specifically, if g is of the form g(x) = h(T(x))-h(x) (a coboundary) with h uniformly bounded then by telescoping the terms in S_n we see that S_n is uniformly bound over n, so when one divides by sqrt(n) the only limit is 0. Thus the limiting distribution is a dirac delta at 0.