I just learned about permutation tests through self-study. According to Wikipedia we have :
(1) "The null hypothesis is that all samples come from the same distribution $H_{0}:F=G$."
and :
(2) "Under the null hypothesis, the distribution of the test statistic is obtained by calculating all possible values of the test statistic under possible rearrangements of the observed data."
I take (2) as synonymous to:
if (1) "$F = G$", then (3) "the sample distribution of the data is exchangeable" , i.e $\text{(1)} \Rightarrow \text{(3)}$.
So, if it's very unlikely for the data at hand to be exchangeable then that means that it would be very unlikely for it to be in a way such that (1) is true. However, according to this post we have :
$$ \text{i.i.d. } \Rightarrow \text{ exchangeability } \Rightarrow \text{marginals identical}.$$
And the converse implications are all wrong.
So here, I'm wondering if $F = G \Leftrightarrow \text{sample data distribution is i.i.d}$, i.e $\text{(1) }\Leftrightarrow \text{i.i.d. }$ as if e.g all of the $X_1,\dots, X_n$ of our sample data distribution are always just implicitly considered independent.
Or, if they're not necessarily independent then we can't really reject (1) with the permutation test because we have : $\text{non exchangeable} \Rightarrow \text{non i.i.d}$. So, a rejection could either "prove" that the $X_k$ are not independent or that they're not distributed the same, without specifying if it's the former, the latter or both.
