On the distribution of values for functions of a large number of variables

It is known that the values of functions that depend on a large number of similar variables are almost constant from the point of view of an observer evaluating their values at random points of the domains of their definition (the nonlinear law of large numbers).
We will recall this phenomenon considering Lipschitz functions on the multidimensional sphere (or on a ball) and we will show that, under certain normalization conditions, the distribution of values of such functions tends to a normal distribution as the number of variables grows.