I'm self-studying large sample theory and I ran into a problem. Can someone help? The image of the problem is here.
Exercise 6.4 Let $X_1,\dots,X_n$ be independent $U(0,1)$ random variables. Find the joint asymptotic distribution of $[nX_{(2)},n(1-X_{(n-1)})]$.
Hint: To find a probability such as $P(a<X_{(2)}<X_{(n)}<b)$, consider the trinomial distribution with parameters $[n;(a,b-a,1-b)]$ and note that the probability in question is the same as the probability that the numbers in the first and third categories are each $\le1$.