Non-linear Functionals Preserving Normal Distribution and Their Asymptotic Normality

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Submitted: 2024-12-17; Published: 2024-12-17

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Abstract

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 one of 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.