In the search, I found the following article in the American Journal of Physics. I found the following paragraph helpful and insightful:

The simplest nontrivial example ofThe image below is taken from a nice site which contains an interactive simulation of the central limit theorem.^{ }universality is given by the law of large numbers (the^{ }central limit theorem) which is crucial in statistical mechanics. In^{ }systems where it can be applied, all the details of^{ }the underlying probability distribution of the constituents of the system^{ }are irrelevant for the cooperative phenomena which are governed by^{ }a Gaussian probability distribution. This drastic reduction of complexity is^{ }precisely what is necessary for physics because it lets us^{ }build effective theories in which only a few couplings are^{ }kept. Renormalizability in statistical field theory is one of the^{ }nontrivial generalizations of the central limit theorem.

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