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

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
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