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

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 $p_\Xi(x,y) = \frac{(a+1)(a+2)}2 |x-y|^{a}1_{[0,1]}(x)1_{[0,1]}(y)$, $a\in(-1,0)$.