Setting
Let $s\in\mathbb{N}$, $E=\{1,2,3,\cdots,s\}$. Also, let $(J,S)=(J_n,S_n)_{n\in\mathbb{N}}$ be a discrete homogeneus Markov renewal chain with semi-Markov kernel $q$. That is, for all $n\in\mathbb{N}$, $J_n\in E$, $S_n\in \mathbb{N}$, and the following property almost surely holds for all $i,j\in E$ and $k\in\mathbb{N}$, being independent of $n$: $$\mathbb{P}(J_{n+1}=j,S_{n+1}-S_n=k| J_1,J_2,\cdots,J_n,S_1,S_2,\cdots ,S_n)=\mathbb{P}(J_{n+1}=j,S_{n+1}-S_n=k|J_n),$$ and thus one can define a semi-Markov kernel $q$ by $$q_{ij}(k)=\mathbb{P}(J_{n+1}=j,S_{n+1}-S_n=k|J_n=i).$$ (Notations were borrrowed from Bardu and Limnios, "Semi-Markov chain and Hidden Semi-Markov Models toward Applications" (Springer))
Now suppose we "observe" the Markov renewal chain for finite time. That is, let $M\in\mathbb{N}$, and we have the following sequence as a data:$$(J_0,S_1,J_1,S_2,\cdots,J_{N(M)-1},S_{N(M)},J_{N(M)}),$$ where $N(M)$ is a natural number satisfying $$S_{N(M)}<M,\quad S_{N(M)+1}\ge M.$$ If we repeat the observation $K(\in\mathbb{N})$ times, then we have $K$ independent sequences, although the length of each sequence varies in general. Then, suppose one would like to estimate the matrix element of $q$ with using a data one has.
Question
I am aware that one can construct empirical estimators, for example for $K=1$, $$\hat{q}_{ij}(k,M)=\frac{T_{ij}(k,M)}{V_i(M)},$$ where $V_i(M)$ ($i\in E$) is a number of visits to the state $i$ and $T_{ij}(k,M)$ ($i,j\in E$) is a number of transitions from the state $i$ to the state $j$ with sojourn time of $k$;$$J_n=i, J_{n+1}=j\ \mathrm{with}\ S_{n+1}-S_{n}=k,$$ and that asymptotic properties of such an estimator as $M\rightarrow\infty$ is studied well in literatures.
However, I have no idea whether one can construct an estimator which is asymptotically consistent as $K\rightarrow\infty$ for fixed $M$ (and if one can, how), nor I am not aware of literatures which provide a clear answer to this problem. (Although in a case of $K>1$, one can construct a similar estimator $$\hat{q}_{ij}(k,M)=\frac{\sum_{\kappa=1}^{K}T_{ij}(k,M;\kappa)}{\sum_{\kappa=1}^{K}V_i(M;\kappa)},$$ where $\kappa$ represents it is $\kappa$th observation, I do not believe this to be asymptotically consistent as $K\rightarrow\infty$). Of course one has no chance of giving an estimate of $q_{ij}(k)$ for $k\ge M$, but how about for $k<M$?
To repeat the question: Can we construct an estimator of semi-Markov kernel, which is asymptotically consistent as $K\rightarrow \infty$ for fixed $M$? If yes, how can we do this? If no, why can't we do this and what can we do to estimate the $q$ at best? Do you know any literatures which provide an answer to these questions?
Any suggestions and partial answer to this question are also gratefully appreciated.