# Higher sample moments of the $n$-consecutive sum of $N$ random variables ($N>n$)

Let $\mathbf{X}$ be an $N$-vector of independent random variables drawn from some distribution with mean $\mu$ and variance $\sigma^2$: $$\mathrm{E}\left[x_1\right]=\cdots=\mathrm{E}\left[x_N\right]=\mu, \mathrm{Var}\left[x_1\right]=\cdots\mathrm{Var}\left[x_N\right]=\sigma^2$$ Let $\mathbf{Y}$ be the vectors of $n$-consecutive sums of $\mathbf{X}$: $$y_i=\sum_{j=i}^{i+n-1}x_j,i=1,\cdots,N-n+1\tag{1}\label{1}$$ or $$\mathbf{Y}=\mathbf{X}\mathbf{M}\tag{2}\label{2}$$ where $$\mathbf{M}=\begin{bmatrix} \left.\begin{matrix} 1& 0&\cdots&0\\ 1& 1&\cdots&0\\ \vdots & \vdots&\ddots&\vdots\\ \end{matrix}\right\}&N-n\\ \left.\begin{matrix} 1& 1 &\cdots&1\\ 0&1&\cdots&1\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1\end{matrix}\right\}&n \end{bmatrix}$$ I have worked out the results of $\mathrm{E}\left[\overline{y_i^r}\right]$, $\mathrm{Var}\left[\overline{y_i^r}\right]$, $\mathrm{E}\left[\overline{\left(y_i-\overline{y_i}\right)^r}\right]$ and $\mathrm{Var}\left[\overline{\left(y_i-\overline{y_i}\right)^r}\right]$ for $r=1,2$, and also $\mathrm{E}\left[\overline {y_i}^2\right]$ and $\mathrm{Var}\left[\overline {y_i}^2\right]$, if useful. I am hindered by the cases of $r=3,4$. Can they be expressed by the known results above?

For example, I see no hope in expanding $\overline{y_i^4}$ by eq. \eqref{1}, neither can I express $\overline{y_i^4}$ into some quadratic forms by eq. \eqref{2}. I found on the Internet some discussions about the maximum of partial sum of random variables (for example, http://msp.org/pjm/1974/52-2/pjm-v52-n2-p27-p.pdf), but no discussions of the sample moments of it. One special characteristic of the current problems is $y_i$ and $y_{i+1}$ are highly overlapped and hence not independent. I know limited mathematical tools but are willing to learn.