Chebyshev polynomials and $G$-distributed functions of $F$-distributed variables

Abstract

We address a more general version of a classic question in probability theory. Suppose $\bf {X \sim N_p({\mu},\Sigma )}$. What functions of $\bf {X}$ also have the $N_p({\mu},\Sigma )$ distribution? For $p = 1$, we give a general result on functions that cannot have this special property. On the other hand, for the $p = 2,3$ cases, we give a family of new nonlinear and non-analytic functions with this property by using the Chebyshev polynomials of the first, second and the third kind. As a consequence, a family of rational functions of a Cauchy-distributed variable are seen to be also Cauchy distributed. Also, with three i.i.d. $N(0,1)$ variables, we provide a family of functions of them each of which is distributed as the symmetric stable law with exponent $\frac{1}{2}$. The article starts with a result with astronomical origin on the reciprocal of the square root of an infinite sum of nonlinear functions of normal variables being also normally distributed; this result, aside from its astronomical interest, illustrates the complexity of functions of normal variables that can also be normally distributed.