A dozen de Finetti-style results in search of a theory. (English) Zbl 0619.60039

Diaconis, Persi; Freedman, David

Ann. Inst. Henri Poincaré, Probab. Stat. 23, Suppl., 397-423 (1987).

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

Keywords:

de Finetti’s theorem; exchangeable; orthogonally invariant; variation distance; uniqueness of mixtures; historical remarks