The Moment Problem for Polynomial Forms in Normal Random Variables

Abstract

Let $Y$ be a random variable defined by a polynomial $p(W)$ of degree $n$ in finitely many normally distributed variables. This paper studies which such variables $Y$ are "determinate," i.e., have probability laws uniquely determined by their moments. Extending results of Berg, which applied to powers of a single normal variable, we prove that (a) $Y$ is determinate if $n = 1, 2$ or if $n = 4$, with the essential support of the law of $Y$ strictly smaller than the real line, and (b) $Y$ is not determinate either if $n$ is odd $\geq 3$ or if $n$ is even $\geq 6$ such that $p(\mathbf{w})$ attains a finite minimum value. Some other polynomials $Y = p(\mathbf{W})$ with even degree $n \geq 4$ are proved not to be determinate.