My question is much more specific than the title: 



> 
> Given a symmetric
>  distribution $\Xi$ on $\mathbb R^2$, when is it possible to construct a sequence $\xi\_1,\xi\_2,\dots$ of random variables such that the joint distribution of any two of them is $\Xi$?
> 
> 
> 


For example, if the pdf of $\Xi$ is decomposable: $p\_\Xi(x,y) = p(x) p(y)$, then one can just take a sequence of independent r.v.'s. 


(To construct certain counterexample related to fractional Brownian motion) I am particularly interested in the pdf $p\_\Xi(x,y) = \frac{(a+1)(a+2)}2 |x-y|^{a}1\_{[0,1]}(x)1\_{[0,1]}(y)$, $a\in(-1,0)$.



When is it possible to construct a joint law from its two-dimensional marginals?
I recently came upon this question in the context of distributions taking values in a finite set, but since yours take values in the compact interval $[0,1]$ I don't think much will go wrong applying the answer to your setting.


Certainly a sufficient condition is that you can construct an exchangeable sequence $\chi\_1,\chi\_2,\ldots$ for which the marginal of $\chi\_1$ and $\chi\_2$ is $\Xi$, or equivalently that $\Xi$ is a mixture of i.i.d. distributions per de Finetti's theorem. It turns out this is also necessary.


To see this suppose that $\Xi$ satisfies the condition you give on its two-variable marginals. Then for any finite $n$ there is a finite sequence of random variables $\xi\_1,\ldots,\xi\_n$ (just take the first $n$ variables of your given sequence) all of whose two-variable marginals are $\Xi$. We can construct a new sequence of random variables $\chi\_1,\ldots,\chi\_n$ by randomly permuting the $\xi\_1,\ldots,\xi\_n$. By linearity the marginal of any two of these will still be $\Xi$.


But the distribution of the $\chi\_1,\ldots,\chi\_n$ is invariant under arbitrary permutations, by symmetry. Using a diagonalization and compactness argument, we get from the existence of such a sequence $\chi\_1,\ldots,\chi\_n$ for all finite $n$ the existence of an exchangeable sequence $\chi\_1,\chi\_2,\ldots$ whose two-variable marginal distribution is $\Xi$.


