Moments of passage time

We will use the matrices introduced above to derive $\mathbf{G}^{(\ell)}$ and $\mathbf{G}^{(\ell)}_r$ for $r=1,2,3$. We first derive those for the repeating part, $\mathbf{G} \equiv \mathbf{G}^{(\hat{\ell})}$ and $\mathbf{G}_r \equiv \mathbf{G}^{(\hat{\ell})}_r$. Then, $\mathbf{G}$ and $\mathbf{G}_r$ are used to derive those for the nonrepeating part, $\mathbf{G}^{(\ell)}$ and $\mathbf{G}^{(\ell)}_r$ for $1\leq \ell<\hat{\ell}$.