Let
\({\mathbb {P}}_{z}(\cdot ):={\mathbb {P}}( \,\cdot \, | \zeta = z)\) and
\({\mathbb {E}}_{z}[\,\cdot \,]:={\mathbb {E}}[\,\cdot \,| \zeta = z]\). Let
\(P^z_{i}\) and
\(\Lambda _{ij}^z\) be the state occupation probabilities and the transition rates that correspond to the censored indicator process
$$\begin{aligned} I^z_i(t) = \mathbb {1}_{\{\zeta =z, J =1, L<t\le R\}} \, I_{i}(t) \end{aligned}$$
and the censored counting process
$$\begin{aligned} N^z_{ij}(t) = \mathbb {1}_{\{\zeta =z, J =1 \}} \big ( N_{ij}(t\wedge R ) - N_{ij}(t\wedge L )\big ) = \int _{(0,t]} \mathbb {1}_{\{\zeta =z, J =1, L<u\le R\}} \mathrm {d}N_{ij} (u) . \end{aligned}$$
By using assumption (b) and the Campbell Theorem, for
\(t\le s\) we can show that
$$\begin{aligned} P_{ij}^z(t) - P_{ij}^z(s)&={\mathbb {E}}[ N_{ij}^z(t) - N_{ij}^z(s) ] \\&= {\mathbb {E}}\bigg [ - \mathbb {1}_{\{ \zeta =z\}} \int _{(t,s]} {\mathbb {E}}[ \mathbb {1}_{\{ J=1, L< u \le R \}}| (Z_t)_{t \ge 0}, \zeta ]\, \mathrm {d}N_{ij} (u) \bigg ]\\&= -{\mathbb {P}}(\zeta =z) \,{\mathbb {E}}_z \bigg [ \int _{(t,s]} {\mathbb {E}}[ \mathbb {1}_{\{ J=1, L< u \le R \}}]\, \mathrm {d}N_{ij} (u) \bigg ]\\&= - {\mathbb {P}}(\zeta =z) \int _{(t,s]}{\mathbb {E}}[ \mathbb {1}_{\{ J=1, L < u \le R \}}] \, \mathrm {d}P_{z,ij}(u) . \end{aligned}$$
On the other hand, assumption (b) implies that
$$\begin{aligned} P_{i}^z(t-) = {\mathbb {E}}[ I^z_{i}(t-) ] = {\mathbb {P}}(\zeta =z) \, {\mathbb {E}}_z [ I_{i}(t-) ] \, {\mathbb {E}}[ \mathbb {1}_{\{ J=1, L < t \le R \}}]. \end{aligned}$$
The latter two equations and assumption (d) yield
$$\begin{aligned} \Lambda _{z,ij} (t) = \Lambda ^z_{ij} (t) \end{aligned}$$
(36)
for
\(t \le s\). Similar calculations show that (
36) holds also for
\(t>s\). For
\(p\in [1,2)\), we define the
p-variation norm as
\(\Vert \cdot \Vert _{[p]}:=\Vert \cdot \Vert _{\infty } + \Vert \cdot \Vert _{V_p}\), where
\(\Vert \cdot \Vert _{\infty }\) is the supremum norm on [0,
T] and
\(\Vert \cdot \Vert _{V_p}\) is the total
p-variation on [0,
T]. According to Theorem 3 in Overgaard [
10], it holds that
$$\begin{aligned} {\mathbb {E}}\Big [ \Vert n^{-1} {\hat{N}}_{\! z,ij} - P_{ij}^z\Vert _{[p]}\Big ] \rightarrow 0 , \quad n \rightarrow \infty , \end{aligned}$$
(37)
for
\(p \in (1,2)\), where
\(P_{ij}^z(t):= {\mathbb {E}}[ N^z_{ij}(t) ]\),
\(t \ge 0\). Because of Eq. (
2), we have
$$\begin{aligned}&|n^{-1} {\hat{I}}_{\! z,i}(t) - P^z_i(t) | \\&\quad \le | n^{-1} {\hat{I}}_{\! z,i}(s) - P^z_i(s) | + \sum _{i,j:i \ne j} |n^{-1} {\hat{N}}_{\! z,ij}(t) - P^z_{ij}(t) | + \sum _{i,j:i \ne j} | n^{-1} {\hat{N}}_{\! z,ij}(s) - P^z_{ij}(s) |. \end{aligned}$$
By using assumption (a), the law of large numbers, dominated convergence and Theorem 3 from Overgaard [
10], we obtain
$$\begin{aligned} {\mathbb {E}}\Big [ \Vert n^{-1} {\hat{I}}_{\! z,i} - P_{i}^z\Vert _{[p]}\Big ] \rightarrow 0, \quad n \rightarrow \infty , \end{aligned}$$
(38)
for
\(p \in (1,2)\). The inequalities
\(\Vert \int _{(0, \cdot ]} g(s) \mathrm {d}f(s) \Vert _{[p]} \le k_p \Vert f\Vert _{[p]} \Vert g\Vert _{[p]}\), see Dudley [
5], and
\(\Vert fg\Vert _{[p]} \le \Vert f\Vert _{[p]} \Vert g\Vert _{[p]}\) imply that
$$\begin{aligned} \bigg \Vert \int _{(0, \cdot ]} (g(s))^{-1} \mathrm {d}f(s) -\int _{(0, \cdot ]}(g'(s))^{-1} \mathrm {d}f'(s)\bigg \Vert _{[p]} \le \frac{\Vert g-g'\Vert _{[p]}}{\Vert g g'\Vert _{[p]}} \Vert f \Vert _{[p]} + \Vert g'\Vert _{[p]} \Vert f-f'\Vert _{[p]}. \end{aligned}$$
(39)
Because of this inequality and (
37), (
38) and assumption (d), we can conclude that
$$\begin{aligned} {\mathbb {E}}\Big [ \Vert {\hat{\Lambda }}_{z,ij} - \Lambda ^z_{ij} \Vert _{[p]}\Big ] \rightarrow 0, \quad n \rightarrow \infty , \end{aligned}$$
(40)
for
\(p \in (1,2)\). Finally, in the latter formula we replace
\(\Lambda ^z_{ij}\) by
\(\Lambda _{z,ij}\), see (
36).
\(\square\)