The cube of a normal distribution is indeterminate. (English) Zbl 0645.60018

Berg, Christian

Ann. Probab. 16, No. 2, 910-913 (1988).

Let \({\mathcal M}\) * be the set of probability measures on the real axis having moments of all orders. Put \(s_ k(\mu)=\int x\) \(k\mu\) (dx), \(k=0,1,..\). for \(\mu\in {\mathcal M}\) *. Two elements \(\mu\),\(\nu\in {\mathcal M}\) are said to be equivalent if \(s_ k(\mu)=s_ k(\nu)\) for all \(k=0,1,...\); \(\mu\in {\mathcal M}*\) is said to be determinate (in the Hamburger sense) if the equivalence class containing \(\mu\) reduces to \(\{\mu\}\), and indeterminate otherwise. It is proved that if X has a normal distribution, then \(X^{2n+1}\) has an indeterminate distribution for \(n\geq 1\). Next, the distribution of \(| X|^{\alpha}\) is determinate for \(0<\alpha \leq 4\) while indeterminate for \(\alpha >4\).

Reviewer: M.Iosifescu

MSC:

60E05	Probability distributions: general theory
44A60	Moment problems

Keywords:

moment problem; moments of all orders; normal distribution