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Zbl 0619.60039
Diaconis, Persi; Freedman, David
A dozen de Finetti-style results in search of a theory.
(English)
[J] Ann. Inst. Henri PoincarĂ©, Probab. Stat. 23, Suppl., 397-423 (1987). ISSN 0246-0203

It is shown that if $\xi =(\xi\sb 1,\xi\sb 2,...,\xi\sb n)$ is uniformly distributed on the surface of the sphere $\{\xi$ :$\sum\sp{n}\sb{i=1}\xi\sp 2\sb i=n\}$ and $1\le k\le n-4$, then the variation distance between the law of $(\xi\sb 1,...,\xi\sb k)$ and the joint law of k independent standard normal variables is less than or equal to $2(k+3)/(n-k-3)$. It follows from this that if a law in ${\bbfR}\sp k$ is orthogonally invariant, then it is within variation distance $2(k+3)/(n-k-3)$ of a mixture of joint laws of i.i.d. centred normals. \par Similar results are proved for the exponential, geometric and Poisson distributions: for example if $(\xi\sb 1,...,\xi\sb n)$ is uniform on the simplex $\{\xi:\xi\sb i\ge 0$ for all i and $\sum\sp{n}\sb{i=1}\xi\sb i=n\}$ then, for $1\le k\le n-2$, the law of $(\xi\sb 1,...,\xi\sb k)$ is within variation distance $2(k+1)/(n-k+1)$ of the joint law of k i.i.d. exponential variables with parameter 1. \par The paper discusses sharpness of bounds, questions of uniqueness of mixtures etc., and concludes with extensive historical remarks.
[F.Papangelou]
MSC 2000:
*60G09 Exchangeability
60J05 Markov processes with discrete parameter
60G10 Stationary processes

Keywords: de Finetti's theorem; exchangeable; orthogonally invariant; variation distance; uniqueness of mixtures; historical remarks

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