Can order statistics be a random samples? 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?

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).
