and that $$\begin{aligned}
{\widetilde}{\bP}^{i}\left(T_{1}>t_{1},\ldots,T_{n}>t_{n} \right)=\prod_{k=1}^{n}\,e^{-q_{k}t_{k}},\quad t_{1},\ldots,t_{n}>0,\end{aligned}$$ which implies that the joint density of $(T_{1},\ldots,T_{n})$ is given by $$\begin{aligned}
\label{eq:densityT}
f_{T_{1},\ldots,T_{n}}(t_{1},\ldots,t_{n})=\prod_{k=1}^{n}q_{k}\,e^{-q_{k}t_{k}},\quad t_{1},\ldots,t_{n}>0.\end{aligned}$$ Combining and , we deduce that $$\begin{aligned}
f_{S_{1},\ldots,S_{n}}(s_{1},\ldots,s_{n})=\prod_{k=1}^{n}q_{k}\,e^{-q_{k}(s_{k}-s_{k-1})},\quad 0=s_{0}<s_{1}<\ldots<s_{n}.\end{aligned}$$

\[thm:ConstBarPi\] For any $i\in\mathbf{E}$, any $0<s_{1}<\ldots<s_{n}$, and any cylinder set $A\in{\widetilde}{\cF}_{\infty}^{Y}$ of the form $$\begin{aligned}
A=\left\{\left(Y_{u_{1}},\ldots,Y_{u_{m}}\right)\in B\right\},\quad 0\leq u_{1}<u_{2}<\ldots<u_{m},\quad B\subseteq\mathbf{E}^{m},\quad m\in\bN,\end{aligned}$$ the limit $$\begin{aligned}
\label{eq:DefBarPiCylSet}
\overline{\bP}^{i}\left(A;s_{1},\ldots,s_{n}\right):=\lim_{\Delta s_{k}\rightarrow 0,\,k=1,\ldots,n}\frac{{\widetilde}{\bP}^{i}\left(A,\,s_{k}<S_{k}\leq s_{k}+\Delta s_{k},\,k=1,\ldots,n\right)}{{\widetilde}{\bP}^{i}\left(s_{k}<S_{k}\leq s_{k}+\Delta s_{k},\,k=1,\ldots,n\right)},\end{aligned}$$ exists, and can be extended to a probability measure $\overline{\bP}^{i}(\cdot\,;s_{1},\ldots,s_{n})$ on $({\widetilde}{\Omega},{\widetilde}{\cF}_{\infty}^{Y})$. Moreover, for any $A\in{\widetilde}{\cF}_{\infty}^{Y}$, the function $\overline{\bP}^{i}(A;\,\ldots)$ is Borel measurable on $\{(s_{1},\ldots,s_{n})\in\bR^{n}\,|\,0<s_{1}<\ldots<s_{n}\}$, and $$\begin{aligned}
\label{eq:TildePiBarPi}
{\widetilde}{\bP}^{i}(A)=\int_{0}^{\infty}\int_{s_{1}}^{\infty}\cdots\int_{s_{n-1}}^{\infty}\overline{\bP}^{i}(A;s_{1},\ldots,s_{n})\prod_{k=1}^{n}\left(q_{k}\,e^{-q_{k}(s_{k}-s_{k-1})}\right)ds_{n}\cdots ds_{2}\,ds_{1}.\end{aligned}$$

In the proof of the theorem we will use the following lemma.

\[lem:ConstBarPiCylSet\] Let us fix $i\in\mathbf{E}$, $0<s_{1}<\ldots<s_{n}$, and let $0=k_{0}<k_{1}<\ldots<k_{n+1}$ be positive integers. In addition, let $0=u_0< u_{1}<\ldots<u_{k_{1}}\leq s_{1}<u_{k_{1}+1}<\ldots<u_{k_{2}}\leq s_{2}<\ldots\leq s_{n}<u_{k_{n}+1}<\ldots<u_{k_{n+1}}$, $i_0=i$ and $i_{1},\ldots,i_{k_{n+1}}\in\mathbf{E}$. Then, for any cylinder set $A\in{\widetilde}{\cF}_{\infty}^{Y}$ of the form $$\begin{aligned}
\label{eq:CylSet}
A=\bigcap_{j=0}^{n}\left\{Y_{u_{k_{j}+1}}=i_{k_{j}+1},\ldots,Y_{u_{k_{j+1}}}\!=i_{k_{j+1}}\right\}\end{aligned}$$ we have $$\begin{aligned}
\lim_{\Delta s_{\ell}\rightarrow 0,\,\ell=1,\ldots,n}&\frac{{\widetilde}{\bP}^{i}\left(A,\,s_{\ell}<S_{\ell}\leq s_{\ell}+\Delta s_{\ell},\,\ell=1,\ldots,n\right)}{{\widetilde}{\bP}^{i}\left(s_{\ell}<S_{\ell}\leq s_{\ell}+\Delta s_{\ell},\,\ell=1,\ldots,n\right)}
=\!\prod_{\ell=0}^{n}\!\left(\!\prod_{m=k_{\ell}+1}^{k_{\ell+1}}\!\!\!\!e^{(u_{m}-u_{m-1})
\mG_{\ell}}\!(i_{m-1},i_{m})\!\right)  \nonumber \\
& \qquad\quad \cdot\sum_{j_{1},\ldots,j_{n}\in\mathbf{E}}\prod_{\ell=1}^{n}e^{(s_{\ell}-u_{k_{\ell}})
\mG_{\ell-1}}\!(i_{k_{\ell}},j_{\ell})e^{(u_{k_{\ell}+1}-s_{\ell})\mG_{\ell}}\!(j_{\ell},i_{k_{\ell}+1}).
\label{eq:BarPiSimCylSet}\end{aligned}$$ In particular, for any $A\in{\widetilde}{\cF}_{\infty}^{Y}$ of the form , the above limit is Borel measurable with respect to $(s_{1},\ldots,s_{n})$ in $\Delta_n:=\{(s_{1},\ldots,s_{n})\in\bR^{n}\,|\,0<s_{1}<\ldots<s_{n}\}$.

For $\ell=1,\ldots,n$ choose $\Delta s_{\ell}>0$ so that, $s_{\ell}+\Delta s_{\ell}\leq u_{k_{\ell}+1}$. Then, $$\begin{aligned}
&{\widetilde}{\bP}^{i}\left(A,\,s_{\ell}<S_{\ell}\leq s_{\ell}+\Delta s_{\ell},\,\ell=1,\ldots,n\right)\\
&\quad ={\widetilde}{\bP}^{i}\left(Y_{u_{k_{\ell}+1}}=i_{k_{\ell}+1},\ldots,Y_{u_{k_{\ell+1}}}=i_{k_{\ell+1}},\,\ell=0,\ldots,n;\,N_{s_{\ell}}=\ell-1,N_{s_{\ell}+\Delta s_{\ell}}=\ell,\,\ell=1,\ldots,n\right)\\
&\quad =\sum_{j_{1},\ldots,j_{n},\,j_{1}',\ldots,j_{n}'\in\mathbf{E}}{\widetilde}{\bP}^{i}\left(Z_{u_{k_{\ell}+1}}=(\ell,i_{k_{\ell}+1}),\,\ldots,\,Z_{u_{k_{\ell+1}}}=(\ell,i_{k_{\ell+1}}),\,\ell=0,\ldots,n;\right.\\
&\qquad\qquad\qquad\qquad\qquad\qquad\,\,\,Z_{s_{\ell}}=(\ell-1,j_{\ell}),\,Z_{s_{\ell}+\Delta s_{\ell}}=(\ell,j_{\ell}'),\,\,\ell=1,\ldots,n\Big)\\
