Consider a random vector $X$ of dimension $L\times 1$ with cumulative distribution function $F$ absolutely continuous. Let $F_1,..., F_L$ denote the marginal cdf's.
Assume that the probability distribution of $X$ is exchangeable, i.e., $$ (\star) \hspace{1cm}F(x_1,..., x_L)=F(x_{\pi(1)},...,x_{\pi(L)}) $$ for any permutation $\pi$ of $\{1,...,L\}$.
Question: are there any sufficient conditions on $F_1,..., F_L$ implying $(\star)$?
(For example, I think it should be that $F_1=...=F_L$ but this is necessary and not sufficient for exchangeability).