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VOL. 45 | 2004 Chebyshev polynomials and $G$-distributed functions of $F$-distributed variables


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.


Published: 1 January 2004
First available in Project Euclid: 28 November 2007

zbMATH: 1268.60014
MathSciNet: MR2126894

Digital Object Identifier: 10.1214/lnms/1196285387

Primary: 05E35 , 60E05 , 85A04

Keywords: analytic , Cauchy , Chebyshev polynomials , normal , one-to-one , Stable law , three term recursion

Rights: Copyright © 2004, Institute of Mathematical Statistics


Vol. 45 • 1 January 2004
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