Assume that we have a sequence of $n$ realizations $x_1, x_2, ..., x_n$ of a i.i.d random variable $X$ 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 $Y_1, ..., Y_{n-k+1}$ where $Y_i$ contains $k-1$ elements of $Y_{i-1}$ (and $Y_{i+1}$).
Further, let $Y_{(l)}$ be $l^{th}$ order statistic of the sequence $Y_1, ..., Y_{n-k+1}$ (i.e. the $l^{th}$ largest $Y^k$) and lets assume that we want to find $F_{Y_{(l)}}(x)$, the cdf of $Y_{(l)}$.
If the $Y_i$ are i.i.d., according to David & Nagarja (2003), $Y_{(l)}$ has the following distribution:
$F_{Y_{(l)}}(x) = \sum_{i = l}^{m} \binom{m}{i}\cdot [F_Y(x)]^{i}\cdot[1-F_Y(X)]^{m-i}, \quad m = n - k+1 \quad (1)$
However, for $k>1$, I understand $Y_{i}$ to be dependent due to the overlapping nature of their construction, in which case (1) does not seem to apply. How do I then define $F_{Y_{(l)}}(x)$? Or do I simply conclude from $X$ being i.i.d. that $Y$ i.i.d. such that (1) still applies?