Distribution of sum of i.i.d random variables

Question

Assume that we have a sequence of $n$ realizations $x_1, x_2, ..., x_n$ of an 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 $Y_{i+1}$).

What is the disribution $F_Y(x)$ of $Y$?

Answer 1

I will denote $Y^k = \sum_{j=i}^k X_j$, as the dependence on $k$ is what matters.

1. As you said, the pdf $f_{Y^k}(x)$ is given by the $k$-fold convolution of $f_X(x)$. You can prove this using the characteristic function of $Y^k$, $\phi_{Y^k}(t) = \langle e^{it Y^k} \rangle_{Y^k} = \langle e^{it \sum_{j=i}^k X_j} \rangle_{X_i, \dots, X_{k+i}} $

2. You don't need to distinguish these two cases, as the pdf of $Y^k$ depends only on $k$ (not on $n$). In fact, you can prove that the distribution of $Y^k$ does not depend on $i$. For large $k$, you have a central limit theorem for $Y^k$.

3. That is true also for finite $n$ (see point $1$).