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In Kingman, J. (1972) On Random Sequences with Spherical Symmetry. Biometrika, 59(2), 492-494., the author states a theorem that, in his words, "has a close resemblance to de Finetti's theorem on exchangeability".

To my best knowledge a sequence of random variables has the property of spherical symmetry if its distribution function is invariant to rotations of its elements, i.e. the distribution function evaluated in $X$ and $ΓX$ is the same given that $Γ$ is any orthogonal $nxn$ matrix.

The only thing I notice is that exchangeability is a property concerning distribution functions invariant with respect to something, i.e. permutations.

Could you please explain how these two concepts are related or different?

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    $\begingroup$ Exchangeability is a generalization of independence. A set of exchangeable random variables have the same joint distribution when the order of the variables are interchange. The distribution doesn't change under permutations of the random variables (invariance). Variables that are statistically independent satisfy this property. You can easily see this in the case of distributions that have densities since the densities factor for independent variables. $\endgroup$ Dec 7 '16 at 0:07
  • $\begingroup$ But how is it related to the concept of spherical symmetry? Is it a broader generalization? $\endgroup$
    – PhDing
    Dec 7 '16 at 8:25
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Spherical symmetry is a special case of exchangeability.

The invariance property for spherically symmetric sequences that you describe is indeed the standard definition of spherical symmetry e.g. see On Ali's Characterization of the Spherical Normal Distribution, Steven F. Arnold and James Lynch, Journal of the Royal Statistical Society. Series B (Methodological) Vol. 44, No. 1 (1982), pp. 49-51.

So, spherically symmetric sequences must be invariant to all orthogonal transformations. However, exchangeable sequences need only be invariant to a subclass of these transformations, namely those transformations representing permutations of the coordinate axes. It follows that all spherically symmetric sequences are exchangeable (and it is this that Kingman uses to prove the theorem in the paper you cite)... but the converse is false.

Example of an exchangeable but non-spherically symmetric sequence. Let $X_1$,$X_2$ be iid standard normal. Let $U$ be independent of the $X_i$, taking value $0$ with probability $0.5$ and taking value $100$ with probability $0.5$. Then the sequence $Y_i:= X_i+U$ is exchangeable but not spherically symmetric. Indeed, most of the mass of the joint distribution of $(Y_1,Y_2)$ surrounds the points $(0,0)$ and $(100,100)$. This is clearly not invariant to rotations.

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    $\begingroup$ Spherical symmetric distributions are relatively restricted, from my knowledge their use is mainly in spatial statistics. But thanks for the nice answer! $\endgroup$
    – Henry.L
    Dec 14 '16 at 2:40

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