The Moment Problem for Polynomial Forms in Normal Random Variables

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 2 3 or if n is even 2 6 such that p(w) attains a finite minimum value. Some other polynomials Y = p(W) with even degree n 2 4 are proved not to be determinate.