Can order statistics be a random samples?

Question

We can say a random sample means that $ X_1, X_2, \ldots , X_n $ are iid from a same distribution. But Can we use a same analogue to order statistics? I mean, Can we convert $ X_1, X_2, \ldots , X_n $ to order statistics $ Y_1, Y_2, \ldots, Y_n $ without losing properties of random sample? As I write this question, it is get in my head that because order statistics are relevant to each other, So they never be a random sample. Is that right? Could you explain more sophisticated answer to me?

Answer 1

If by "random sample" you mean an iid collection of random variables, then no, the order statistics do not constitute a random sample. In particular, the order statistics are not identical because you are guaranteed that $$Y_1 \leq Y_2 \leq \ldots \leq Y_n$$

Moreover, if you assume that the $X_i$'s admit densities, then you can argue that the $Y_i$'s are not independent by observing that their joint densities do not factor into the product of their marginal densities (e.g. see here).