&\quad =\sum_{j_{1},\ldots,j_{n},\,j_{1}',\ldots,j_{n}'\in\mathbf{E}}\!\left[\prod_{\ell=0}^{n}\left(\prod_{m=k_{\ell}+1}^{k_{\ell+1}}\!e^{(u_{m}-u_{m-1}){\widetilde}{\mG}}\!\left((\ell,i_{m-1}),(\ell,i_{m})\right)\right)\right]\left(\prod_{\ell=1}^{n}e^{\Delta s_{\ell}{\widetilde}{\mG}}\!\left((\ell-1,j_{\ell}),(\ell,j_{\ell}')\right)\right)\\
&\qquad\qquad\qquad\qquad\quad\cdot\left(\prod_{\ell=1}^{n}e^{(s_{\ell}-u_{k_{\ell}}){\widetilde}{\mG}}\left((\ell-1,i_{k_{\ell}}),(\ell-1,j_{\ell})\right)e^{(u_{k_{\ell}+1}-s_{\ell}-\Delta s_{\ell}){\widetilde}{\mG}}\left((\ell,j_{\ell}'),(\ell,i_{k_{\ell}+1})\right)\right).\end{aligned}$$ In the above summation, the first product in the brackets provides the transition probabilities of the evolutions of $Z$ between the times $u_{k_{\ell}}$ and $u_{k_{\ell+1}}$, $\ell=0,\ldots,n$, the second product gives the transition probabilities of the evolutions of $Z$ between the times $s_{\ell}$ and $s_{\ell}+\Delta s_{\ell}$, for each $\ell=1,\ldots,n$, and the third product denotes the transition probabilities of the evolutions of $Z$ between the times $u_{k_{\ell}}$ and $s_{\ell}$, and between the times $s_{\ell}+\Delta s_{\ell}$ and $u_{k_{\ell}+1}$, for each $\ell=1,\ldots,n$.

Next, for each $\ell=1,\ldots,n$, $$\begin{aligned}
\lim_{\Delta s_{\ell}\rightarrow 0}\frac{1}{\Delta s_{\ell}}\,e^{\Delta s_{\ell}{\widetilde}{\mG}}\left((\ell-1,j_{\ell}),(\ell,j_{\ell}')\right)={\widetilde}{\mG}\left((\ell-1,j_{\ell}),(\ell,j_{\ell}')\right)=\left\{\begin{array}{ll} q_{\ell}, &\text{if }\,j_{\ell}=j_{\ell}', \\ 0, &\text{otherwise}. \end{array}\right.\end{aligned}$$ Hence, $$\begin{aligned}
&\lim_{\Delta s_{\ell}\rightarrow 0,\,\ell=1,\ldots,n}\frac{1}{\Delta s_{1}\cdots\Delta s_{n}}\,{\widetilde}{\bP}^{i}\left(A,\,s_{\ell}<S_{\ell}\leq s_{\ell}+\Delta s_{\ell},\,\ell=1,\ldots,n\right)\nonumber\\
&\quad =\prod_{\ell=0}^{n}\left(\prod_{m=k_{\ell}+1}^{k_{\ell+1}}e^{(u_{m}-u_{m-1}){\widetilde}{\mG}}\!\left((\ell,i_{m-1}),(\ell,i_{m})\right)\right)\nonumber\\
\label{eq:LimProbASell} &\qquad\cdot\sum_{j_{1},\ldots,j_{n}\in\mathbf{E}}\,\prod_{\ell=1}^{n}\left(q_{\ell}\,e^{(s_{\ell}-u_{k_{\ell}}){\widetilde}{\mG}}\left((\ell-1,i_{k_{\ell}}),(\ell-1,j_{\ell})\right)e^{(u_{k_{\ell}+1}-s_{\ell}){\widetilde}{\mG}}\left((\ell,j_{\ell}),(\ell,i_{k_{\ell}+1})\right)\right).\end{aligned}$$ Note that, for any $j_{1},j_{2}\in\mathbf{E}$, and any $k\in\mathbb{N}$, $$\begin{aligned}
{\widetilde}{\mG}^{k}\left((\ell,j_{1}),(\ell,j_{2})\right)&=(\mG_{\ell}-q_{\ell+1}\mI)^{k}(j_{1},j_{2}),\quad\ell=0,\ldots,n-1,\\
{\widetilde}{\mG}^{k}\left((n,j_{1}),(n,j_{2})\right)&=\mG_{n}^{k}(j_{1},j_{2}),\end{aligned}$$ so that, for $t\geq 0$, we have $$\begin{aligned}
e^{t\,{\widetilde}{\mG}}\left((\ell,j_{1}),(\ell,j_{2})\right)&=e^{t\,(\mG_{\ell}-q_{\ell+1}\mI)}(j_{1},j_{2})=e^{-q_{\ell+1}t}\,e^{t\,\mG_{\ell}}(j_{1},j_{2}),\quad\ell=0,\ldots,n-1,\\
e^{t\,{\widetilde}{\mG}}\left((n,j_{1}),(n,j_{2})\right)&=e^{t\,\mG_{n}}(j_{1},j_{2}).\end{aligned}$$ This, together with , implies that $$\begin{aligned}
&\lim_{\Delta s_{\ell}\rightarrow 0,\,\ell=1,\ldots,n}\frac{1}{\Delta s_{1}\cdots\Delta s_{n}}\,{\widetilde}{\bP}^{i}\left(A,\,s_{\ell}<S_{\ell}\leq s_{\ell}+\Delta s_{\ell},\,\ell=1,\ldots,n\right)\\
&\quad =e^{-\sum_{\ell=1}^{n}q_{\ell}(u_{k_{\ell}}-u_{k_{\ell-1}})}\cdot\prod_{\ell=0}^{n}\left(\prod_{m=k_{\ell}+1}^{k_{\ell+1}}e^{(u_{m}-u_{m-1})\mG_{\ell}}(i_{m-1},i_{m})\right)\\ &\qquad\,\,e^{-\sum_{\ell=1}^{n}q_{\ell}(s_{\ell}-u_{k_{\ell}})}e^{-\sum_{\ell=1}^{n-1}q_{\ell}(u_{k_{\ell}+1}-s_{\ell})}\!\!\!\sum_{j_{1},\ldots,j_{n}\in\mathbf{E}}\prod_{\ell=1}^{n}\!\left(q_{\ell}e^{(s_{\ell}-u_{k_{\ell}})\mG_{\ell-1}}(i_{k_{\ell}},j_{\ell})e^{(u_{k_{\ell}+1}-s_{\ell})\mG_{\ell}}(j_{\ell},i_{k_{\ell}+1})\right)\\
&\quad =e^{-\sum_{\ell=1}^{n}q_{\ell}(s_{\ell}-s_{\ell-1})}\cdot\prod_{\ell=0}^{n}\left(\prod_{m=k_{\ell}+1}^{k_{\ell+1}}e^{(u_{m}-u_{m-1})\mG_{\ell}}(i_{m-1},i_{m})\right)\\
&\qquad\,\cdot\sum_{j_{1},\ldots,j_{n}\in\mathbf{E}}\,\prod_{\ell=1}^{n}\left(q_{\ell}\,e^{(s_{\ell}-u_{k_{\ell}})\mG_{\ell-1}}(i_{k_{\ell}},j_{\ell})\,e^{(u_{k_{\ell}+1}-s_{\ell})\mG_{\ell}}(j_{\ell},i_{k_{\ell}+1})\right)\\
&\quad =\left(\prod_{\ell=1}^{n}q_{\ell}\,e^{-q_{\ell}(s_{\ell}-s_{\ell-1})}\right)\cdot\left[\prod_{\ell=0}^{n}\left(\prod_{m=k_{\ell}+1}^{k_{\ell+1}}e^{(u_{m}-u_{m-1})\mG_{\ell}}(i_{m-1},i_{m})\right)\right]\\
&\qquad\,\,\cdot\sum_{j_{1},\ldots,j_{n}\in\mathbf{E}}\,\prod_{\ell=1}^{n}\left(e^{(s_{\ell}-u_{k_{\ell}})\mG_{\ell-1}}(i_{k_{\ell}},j_{\ell})\,e^{(u_{k_{\ell}+1}-s_{\ell})\mG_{\ell}}(j_{\ell},i_{k_{\ell}+1})\right).\end{aligned}$$ Finally, in view of the above and the fact that $$\begin{aligned}
\label{eq:nice-fact}
\lim_{\Delta s_{\ell}\rightarrow 0,\,\ell=1,\ldots,n}\frac{1}{\Delta s_{1}\cdots\Delta s_{n}}\,{\widetilde}{\bP}^{i}\left(s_{\ell}<S_{\ell}\leq s_{\ell}+\Delta s_{\ell},\,\ell=1,\ldots,n\right)=\prod_{\ell=1}^{n}q_{\ell}\,e^{-q_{\ell}(s_{\ell}-s_{\ell-1})},\end{aligned}$$ we obtain . The proof is complete.

We are now ready to prove Theorem \[thm:ConstBarPi\].

Let $\cC$ be the collection of all cylinder sets 