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A dozen de Finetti-style results in search of a theory. (English) Zbl 0619.60039
It is shown that if $$\xi =(\xi_ 1,\xi_ 2,...,\xi_ n)$$ is uniformly distributed on the surface of the sphere $$\{\xi$$ :$$\sum^{n}_{i=1}\xi^ 2_ i=n\}$$ and $$1\leq k\leq n-4$$, then the variation distance between the law of $$(\xi_ 1,...,\xi_ k)$$ 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 $${\mathbb{R}}^ k$$ 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.
Similar results are proved for the exponential, geometric and Poisson distributions: for example if $$(\xi_ 1,...,\xi_ n)$$ is uniform on the simplex $$\{\xi:\xi_ i\geq 0$$ for all i and $$\sum^{n}_{i=1}\xi_ i=n\}$$ then, for $$1\leq k\leq n-2$$, the law of $$(\xi_ 1,...,\xi_ k)$$ is within variation distance $$2(k+1)/(n-k+1)$$ of the joint law of k i.i.d. exponential variables with parameter 1.
The paper discusses sharpness of bounds, questions of uniqueness of mixtures etc., and concludes with extensive historical remarks.
Reviewer: F.Papangelou

##### MSC:
 60G09 Exchangeability for stochastic processes 60J05 Discrete-time Markov processes on general state spaces 60G10 Stationary stochastic processes
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