THE MOMENT PROBLEM FOR POLYNOMIAL FORMS IN NORMAL RANDOM-VARIABLES

• 成果类型: Article
• 署名作者: SLUD, EV
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989017
• 发表日期: 1993
• 页码: 2200-2214
• 关键词: FUNCTIONALS

摘要

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