Chebyshev Polynomials and G-Distributed Functions of F-Distributed Variables

We address a more general version of a classic question in probability theory. Suppose X∼Np(μ,Σ). What functions of X also have the Np(μ,Σ) 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 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.