Suppose that we have a sequence of $n$ i.i.d. random variates $X_1, X_2, ..., X_n$ with cdf $F_X(x)$ and pdf $f_X(x)$. Now define $Y$ as the sum of $k$ consecutive realizations of $X$
$Y_i = \sum_{j=i}^{k} X_j$
which generates the sequence $Y_1, ..., Y_{n-k+1}$ of correlated random variates. $Y_i$ contains $k-1$ elements of $Y_{i-1}$ (and $Y_{i+1}$).
Consider now the order statistics $Y_{(1)}, ..., Y_{(n-k+1)}$.
Is there a way to define the distribution $F_{(i)}(y) = P[Y_{(i)} \leq y]$ of the $i^{th}$ order statistic $ Y_{(i)}$?
Any answer or literature reference is highly appreciated.