Suppose that we have a sequence of $n$ i.i.d. random variables $X_1, X_2, ..., X_n$ with cdf $F_X(x)$ and pdf $f_X(x)$.
Now suppose that there is another sequence of random variables $Y_1, ..., Y_{n-k+1}$ where $Y_i$ is defined as $Y_i = \sum_{j=i}^{k} X_j$ and thus contains $k-1$ elements of $Y_{i-1}$ (and $Y_{i+1}$).
I want to find the distribution of $Y$ but I am not sure how to (as a first step) show or refute the independent and/or identical distribution of $Y$. How can we do so and how can we find the distribution $F_Y(y)$?