Non–linear functionals preserving normal distribution and their asymptotic normality

We introduce sufficiently wide classes of non-linear functionals preserving normal (Gaussian) distribution and establish various conditions under which a sequence of such functionals is asymptotically normal.
As a  consequence, we obtain a generalization and sharpening of known results on the central limit theorem for weighted sums (linear functionals) of independent random variables.

Invited by Sylvie Roelly and Pierre Houdebert