$N$ point moving average filters in state space
Yes sure they are LTI. Let $A$ be the $(L-1)\times (L-1)$ shift matrix 
$$
A := \begin{pmatrix}0 &amp; 1 &amp; 0 &amp;&amp; \dots &amp; 0\\0 &amp; 0 &amp; 1 &amp; 0 &amp;\dots &amp;0\\\vdots &amp;&amp;&amp; \ddots &amp;&amp; 0\\0&amp;&amp;\dots&amp;&amp;1&amp;0\\0 &amp;&amp;\dots&amp;&amp;0&amp;1\\0&amp;0&amp;\dots&amp;0&amp;0&amp;0\end{pmatrix}
$$
and 
$$
B = \begin{pmatrix}0\\0\\\vdots\\0\\1\end{pmatrix},\qquad C=\begin{pmatrix}1/L&amp;1/L&amp;\dots&amp;1/L\end{pmatrix},\quad D=1/L
$$
Call $z$ the state variables, then a state space model is
\begin{align*}z(k+1) &amp;= Az(k) + Bx(k)\\y(k) &amp;= Cz(k)+Dx(k)\end{align*}
Infact one has
\begin{align*}
z_1(k) &amp;= z_2(k-1) = z_3(k-2) = ... = z_{L-1}(k-L) = x(k-(L-1))\\
z_2(k) &amp;= x(k-(L-2))\\
&amp;\vdots\\
z_i(k) &amp;= x(k-(L-i))\\
&amp;\vdots\\
z_{L-1} &amp;= x(k-1)
\end{align*}
thus
\begin{align*}
y(k) &amp;= \dfrac{1}{L}\left(\sum_{i=1}^{L-1} x(k-L+i) + x(k)\right)= \dfrac{1}{L}\left(\sum_{m=1}^{L-1} x(k-m) + x(k)\right) \\&amp;= \dfrac{1}{L}\sum_{m=0}^{L-1}x(k-m)
\end{align*}