let $X_1, X_2,\dots X_n$ denote a sequence of $n$ iid random variables with the first $k$ moments of $F_X$ exist. Under what conditions (if at all) do the first $k$ moments of the random variable $X_{\text{max}}=\max\{\,X_1,\ldots,X_n\,\}$ exist?
I want to know this since I would like to apply the CLT to the mean of $M$ such maximums and before I do that, I need to verify that the moments of the variables used to construct the mean have such finite $k$ first moments.