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1. Department of Mathematical Sciences, National Chengchi University, Taipei City, 116011, Taiwan

   Sheng-Hsien Chang, Chwan-Chin Song & Thomas J. Jiang

2. Institute of Statistics, National University of Kaohsiung, Kaohsiung, 811726, Taiwan

   Kun-Lin Kuo

Correspondence to Kun-Lin Kuo.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

1. (i)

   State \(\varvec{u}\) is an absorbing state in \(T^{(12)}\) means that \(1=t_{\varvec{uu}}^{(12)}=a_{\varvec{u}}b_{\varvec{u}}\) which is equivalent to \(1=b_{\varvec{u}}a_{\varvec{u}}=t_{\varvec{uu}}^{(21)}\), i.e., \(\varvec{u}\) is an absorbing state in \(T^{(21)}\).

2. (ii)

   For all \(\varvec{v}\in \Omega \), we have \(t_{\varvec{vu}}^{(21)}=b_{v_1u_2}a_{\varvec{u}}=0\). Hence, \(\varvec{u}\) is a transition state in \(T^{(21)}\).

3. (iii)

   The proof of (iii) is similar to (ii) and we omit it.

\(\square \)

Without loss of generality, consider \(i=1\) and \(j=2\). Assume that \(\varvec{v}\) is a transient state in \(T^{(21)}\). It gives that \(\varvec{v}{\mathop {\dashrightarrow }\limits ^{21}} \varvec{w}\) for some recurrent state \(\varvec{w}\) in \(T^{(21)}\). Because every row of B contains at least a non-zero entry, there is a state \((w_1,y_2)\in N^{(2)}\) such that \(\varvec{u}{\mathop {\longrightarrow }\limits ^{1}}\varvec{v}{\mathop {\dashrightarrow }\limits ^{21}} \varvec{w}{\mathop {\longrightarrow }\limits ^{2}}(w_1,y_2)\), i.e., \(\varvec{u}{\mathop {\dashrightarrow }\limits ^{12}}(w_1,y_2)\). This implies that both \(\varvec{u}\) and \((w_1,y_2)\) are in the same recurrent class in \(T^{(12)}\) and we then have \((w_1,y_2){\mathop {\dashrightarrow }\limits ^{12}} \varvec{u}\). From \(\varvec{w}{\mathop {\longrightarrow }\limits ^{2}}(w_1,y_2){\mathop {\dashrightarrow }\limits ^{12}} \varvec{u}{\mathop {\longrightarrow }\limits ^{1}}\varvec{v}\), we are given \(\varvec{w}{\mathop {\dashrightarrow }\limits ^{21}}\varvec{v}\). It follows that \(\varvec{v}\) is a recurrent state in \(T^{(21)}\). This contradiction ends the proof. \(\square \)

Consider \(i=1\) and \(j=2\). Fix k and let \(D=\{\varvec{v}: \varvec{u}\in C^{(12)}_{k},\varvec{u}{\mathop {\longrightarrow }\limits ^{1}}\varvec{v}\}\). First, we claim that \(D\ne \emptyset \). For \(\varvec{u}\in C^{(12)}_{k}\), since every column of A contains at least a non-zero entry, there is a state \((v_1,u_2)\in N^{(1)}\) such that \(\varvec{u}{\mathop {\longrightarrow }\limits ^{1}}(v_1,u_2)\), i.e., \((v_1,u_2)\in D\).

Next, D consists of recurrent states only by Lemma 1. Now, we show that all members of D are reachable from each other. For \(\varvec{v}\in D\), there is a \(\varvec{u}\in C^{(12)}_{k}\) such that \(\varvec{u}{\mathop {\longrightarrow }\limits ^{1}}\varvec{v}\). Because every row of B contains at least a non-zero entry, there exists \((v_1,w_2)\in N^{(2)}\) such that \(\varvec{u}{\mathop {\longrightarrow }\limits ^{1}}\varvec{v}{\mathop {\longrightarrow }\limits ^{2}}(v_1,w_2)\). Equivalently, \(\varvec{u}{\mathop {\longrightarrow }\limits ^{12}}(v_1,w_2)\) and then \((v_1,w_2)\in C^{(12)}_{k}\). For any another \(\varvec{v}^*\in D\), there exists \(\varvec{u}^*\in C^{(12)}_{k}\) such that \(\varvec{u}^*{\mathop {\longrightarrow }\limits ^{1}}\varvec{v}^*\). Now, we have \(\varvec{v}{\mathop {\longrightarrow }\limits ^{2}}(v_1,w_2){\mathop {\dashrightarrow }\limits ^{12}}\varvec{u}^*{\mathop {\longrightarrow }\limits ^{1}}\varvec{v}^*\). That is, \(\varvec{v}{\mathop {\dashrightarrow }\limits ^{21}}\varvec{v}^*\).

Finally, we claim that D is closed. Suppose that \(\varvec{v}\in D\) and \(\varvec{v}{\mathop {\dashrightarrow }\limits ^{21}}\varvec{v}^*\). Then there exists \(\varvec{u}\in C^{(12)}_{k}\) such that \(\varvec{u}{\mathop {\longrightarrow }\limits ^{1}}\varvec{v}{\mathop {\dashrightarrow }\limits ^{21}}\varvec{v}^*\) or, equivalently, \(\varvec{u}{\mathop {\dashrightarrow }\limits ^{12}}\varvec{w}{\mathop {\longrightarrow }\limits ^{1}}\varvec{v}^*\) for some \(\varvec{w}\in N^{(2)}\). This shows that \(\varvec{w}\in C^{(12)}_{k}\) and we then have \(\varvec{v}^*\in D\). As a result, D is a recurrent class in \(T^{(21)}\). \(\square \)

1. (i)

   Since \(C^{(21)}_{k}=\{\varvec{v}: \varvec{u}\in C^{(12)}_{k},\varvec{u}{\mathop {\longrightarrow }\limits ^{1}}\varvec{v}\}\), we have \((C^{(12)}_{k})_{X_2}= (C^{(21)}_{k})_{X_2}\). In addition, \((C^{(12)}_{k})_{X_1}= (C^{(21)}_{k})_{X_1}\) because \(C^{(12)}_{k}=\{\varvec{v}: \varvec{u}\in C^{(21)}_{k},\varvec{u}{\mathop {\longrightarrow }\limits ^{2}}\varvec{v}\}\).

2. (ii)

   Consider \(i=1\) and \(j=2\). Assume that \(u_1\in (C^{(12)}_{k})_{X_1}\cap (C^{(12)}_{k^*})_{X_1}\), there must exist two different states such that \((u_1,u_2)\in C^{(12)}_{k}\) and \((u_1,u_2^*)\in C^{(12)}_{k^*}\). By Theorem 1, we can find \((u_1,w_2)\in C^{(21)}_{k}\) such that \((u_1,w_2){\mathop {\longrightarrow }\limits ^{2}}(u_1,u_2)\). However, we also have \((u_1,w_2){\mathop {\longrightarrow }\limits ^{2}}(u_1,u_2^*)\) because \(t^{(2)}_{(u_1,w_2)(u_1,u_2^*)}=b_{u_1u_2^*}>0\). By Theorem 1, this leads to \(k=k^*\), which overthrows the beginning assumption. On the other hand, suppose that \(u_2\in (C^{(12)}_{k})_{X_2}\cap (C^{(12)}_{k^*})_{X_2}\), we have \((u_1,u_2)\in C^{(12)}_{k}\) and \((u_1^*,u_2)\in C^{(12)}_{k^*}\) for some \(u_1\) and \(u_1^*\). Because every column of A and every row of B contain non-zero entries, there exist \((w_1,u_2)\in N^{(1)}\) and \((w_1,w_2)\in N^{(2)}\) such that \((u_1,u_2){\mathop {\longrightarrow }\limits ^{1}}(w_1,u_2){\mathop {\longrightarrow }\limits ^{2}}(w_1,w_2)\) and \((u_1^*,u_2){\mathop {\longrightarrow }\limits ^{1}}(w_1,u_2){\mathop {\longrightarrow }\limits ^{2}}(w_1,w_2)\). That is, \((u_1,u_2){\mathop {\longrightarrow }\limits ^{12}}(w_1,w_2)\) and \((u_1^*,u_2){\mathop {\longrightarrow }\limits ^{12}}(w_1,w_2)\). Therefore, \((w_1,w_2)\) belongs to \(C^{(12)}_{k}\) and \(C^{(12)}_{k^*}\) simultaneously. This is a contradiction. \(\square \)

Consider \(i=1\) and \(j=2\). By the definition of \(T^{(1)}\), it can be shown that the support of \(\widetilde{\varvec{\pi }}_k^{(12)} T^{(1)}\) is given by \(\{\varvec{v} \in N^{(1)}: \varvec{u}\in C^{(12)}_{k},\varvec{u}{\mathop {\longrightarrow }\limits ^{1}}\varvec{v}\}\), which is just \(C_k^{(21)}\). Since

it shows that \(\widetilde{\varvec{\pi }}_k^{(12)}T^{(1)}\) is an augmented vector of \(\varvec{\pi }_k^{(21)}\). However, the augmented vector of \(\varvec{\pi }_k^{(21)}\), \(\widetilde{\varvec{\pi }}_k^{(21)}\), is unique. Therefore, \(\widetilde{\varvec{\pi }}_k^{(12)}T^{(1)}=\widetilde{\varvec{\pi }}_k^{(21)}\) and \(\widetilde{\varvec{\pi }}_k^{(12)} T^{(2)}=\widetilde{\varvec{\pi }}_k^{(12)}T^{(12)}T^{(2)}=\widetilde{\varvec{\pi }}_k^{(12)}T^{(12)}=\widetilde{\varvec{\pi }}_k^{(12)}\). The rest parts can be argued analogously. \(\square \)

1. (i)

   Consider \(i=1\) and \(j=2\). For \(\varvec{\pi }^{(12)} \in \mathcal {V}^{(12)}\), let \(\varvec{\pi }^{(12)}\) be decomposed as \(\varvec{\pi }^{(12)}=(\varvec{0}, \varvec{\pi }_1,\ldots ,\varvec{\pi }_L)\), in which zero vector corresponds to the transient class \(C_0^{(12)}\), and sub-vector \(\varvec{\pi }_k\) corresponds to the recurrent class \(C_k^{(12)}\) for \(k=1,\dots ,L\). From \(\varvec{\pi }^{(12)}T^{(12)}=\varvec{\pi }^{(12)}\), we have \(\varvec{\pi }_k T_k^{(12)}=\varvec{\pi }_k\) for \(k=1,\ldots ,L\). Since \(\varvec{\pi }_k^{(12)}\) is the unique positive stationary probability vector for \(T_k^{(12)}\), it gives \(0 \le \alpha _k\le 1\) so that \(\varvec{\pi }_k=\alpha _k \varvec{\pi }_k^{(12)}\) for \(k=1,\ldots ,L\) and \(\sum _{k=1}^{L} \alpha _k=1\). Consequently, \(\varvec{\pi }^{(12)}=(\varvec{0}, \alpha _1\varvec{\pi }_1^{(12)},\ldots ,\alpha _L\varvec{\pi }_L^{(12)})=\sum _{k=1}^{L}\alpha _k \widetilde{\varvec{\pi }}_k^{(12)}\).

2. (ii)

   It follows immediately from Theorem 3.

3. (iii)

   From Theorem 3, \(\{\widetilde{\varvec{\pi }}_k^{(12)}\}_{k=1}^L\) and \(\{\widetilde{\varvec{\pi }}_k^{(12)}\}_{k=1}^L\) have a one-to-one correspondence, from which a bijective mapping from \(\mathcal {V}^{(12)}\) to \(\mathcal {V}^{(21)}\) is induced accordingly.

4. (iv)

   The results can be shown by (ii) and Proposition 1(i). \(\square \)

1. (i)

   It follows from the well-known results concerning limits of reducible Markov chains (see, for example, Meyer (2000, p. 698)).

2. (ii)

   According to (A2) and (A3), \((T_k^{(12)})^{\infty }\) exists if and only if there exists a positive integer m such that \((T_k^{(12)})^m>0\), i.e., \(\varvec{u}{\mathop {\dashrightarrow }\limits ^{12}}\varvec{v}\) with m steps for all \(\varvec{u},\varvec{v}\in C^{(12)}_{k}\). Now, for all \(\varvec{u}^*,\varvec{v}^*\in C^{(21)}_{k}\), by Theorem 1 and the property of the conditional probability matrix B, there exist \(\varvec{v}\in C^{(12)}_{k}\) and a positive entry, say \(b_{\varvec{u}}\), in the same row with \(b_{\varvec{u}^*}\) such that \(\varvec{v}{\mathop {\longrightarrow }\limits ^{1}} \varvec{v}^*\) and \(\varvec{u}^*{\mathop {\longrightarrow }\limits ^{2}} \varvec{u}\). Clearly, \(\varvec{u}^*{\mathop {\dashrightarrow }\limits ^{21}}\varvec{v}^*\) with \(m+1\) steps because \(\varvec{u}^*{\mathop {\longrightarrow }\limits ^{2}} \varvec{u}{\mathop {\dashrightarrow }\limits ^{12}}\varvec{v}{\mathop {\longrightarrow }\limits ^{1}} \varvec{v}^*\). This proves \((T_k^{(12)})^{m+1}>0\). Since \(T_k^{(21)}\) is irreducible and primitive, hence, \((T_k^{(21)})^{\infty }\) exists. The converse can be proved in the same manner.

3. (iii)

   By (i) and (ii). \(\square \)

Without loss of generality, consider \(i=1\) and \(j=2\).

1. (i)

   For \(\varvec{u}\in N\), we have \(t_{\varvec{u}\varvec{u}}^{(1)}=a_{\varvec{u}}>0\), i.e., \(\varvec{u}{\mathop {\longrightarrow }\limits ^{1}} \varvec{u}\).

2. (ii)

   According to (i), we have \(\varvec{u}{\mathop {\longrightarrow }\limits ^{1}}\varvec{u}{\mathop {\longrightarrow }\limits ^{2}} \varvec{u}\), that is, \(\varvec{u}{\mathop {\longrightarrow }\limits ^{12}} \varvec{u}\).

3. (iii)

   Suppose that \(\varvec{u}{\mathop {\longrightarrow }\limits ^{1}} \varvec{v}\). Then \(u_2=v_2\), which implies \(t_{\varvec{v}\varvec{u}}^{(1)}=a_{\varvec{u}}>0\), i.e., \(\varvec{v}{\mathop {\longrightarrow }\limits ^{1}} \varvec{u}\).

4. (iv)

   \(\varvec{u}{\mathop {\longrightarrow }\limits ^{ij}} \varvec{v}\) follows from \(\varvec{u}{\mathop {\longrightarrow }\limits ^{i}} \varvec{v}{\mathop {\longrightarrow }\limits ^{j}} \varvec{v}\), and \(\varvec{u}{\mathop {\longrightarrow }\limits ^{ji}} \varvec{v}\) follows from \(\varvec{u}{\mathop {\longrightarrow }\limits ^{j}} \varvec{u}{\mathop {\longrightarrow }\limits ^{i}} \varvec{v}\).

5. (v)

   \(\varvec{u}{\mathop {\longrightarrow }\limits ^{12}} \varvec{v}\) implies that \(\varvec{u}{\mathop {\longrightarrow }\limits ^{1}}(v_1,u_2){\mathop {\longrightarrow }\limits ^{2}} \varvec{v}\). By (i) and (iii), we have \(\varvec{v}{\mathop {\longrightarrow }\limits ^{1}}\varvec{v}{\mathop {\longrightarrow }\limits ^{2}}(v_1,u_2){\mathop {\longrightarrow }\limits ^{1}} \varvec{u}{\mathop {\longrightarrow }\limits ^{2}}\varvec{u}\), i.e., \(\varvec{v}{\mathop {\dashrightarrow }\limits ^{12}}\varvec{u}\).

6. (vi)

   By (v). \(\square \)

1. (i)

   Without loss of generality, consider \(i=1\) and \(j=2\). Suppose that \(\varvec{u}\in N\cap C_0^{(12)}\). Then \(\varvec{u}\) is a is a transient state in \(T^{(12)}\). It gives that \(\varvec{u}{\mathop {\dashrightarrow }\limits ^{12}} \varvec{v}\) for some recurrent state \(\varvec{v}\) in \(T^{(12)}\). Because \(\varvec{u},\varvec{v}\in N\) and \(\varvec{u}{\mathop {\dashrightarrow }\limits ^{12}} \varvec{v}\), by Lemma 3(vi), we have \(\varvec{v}{\mathop {\dashrightarrow }\limits ^{12}} \varvec{u}\), which implies that \(\varvec{u}\) is a recurrent state in \(T^{(12)}\). This is a contradiction.

2. (ii)

   Form (i), we obtain \(C_0^{(12)}=C_0^{(21)}\). Now, we consider \(1\le k\le L\). Suppose that \(\varvec{u}\in C_k^{(12)}\). By Theorem 1, there is a \(\varvec{v}\in C_k^{(21)}\) such that \(\varvec{v}{\mathop {\longrightarrow }\limits ^{2}}\varvec{u}\). According to Lemma 3(i), we have \(\varvec{v}{\mathop {\longrightarrow }\limits ^{2}}\varvec{u}{\mathop {\longrightarrow }\limits ^{1}}\varvec{u}\), i.e., \(\varvec{v}{\mathop {\longrightarrow }\limits ^{21}}\varvec{u}\). This implies that \(\varvec{u}\in C_k^{(21)}\). Hence, \(C_k^{(12)}\subseteq C_k^{(21)}\). By the same token, we can show that \(C_k^{(21)}\subseteq C_k^{(12)}\). \(\square \)

1. (i)

   By Lemma 3 and Theorem 6.

2. (ii)

   Since \(T_k^{(ij)}\) is irreducible and has positive diagonal entries, \(T_k^{(ij)}\) is primitive and then \((T_k^{(ij)})^\infty =\varvec{1}^{\prime } \varvec{\pi }_k^{(ij)}\). Moreover, \((T^{(ij)})^\infty \) exists by Theorem 5(i). \(\square \)

Consider \(i=1\). Let \(\varvec{\omega }=\varvec{\pi } T^{(1)}\). Suppose \((i,j) \in E\). Since \(E \subseteq N^{(1)}\), we have \(a_{ij}>0\) and \(0<\pi _{ij}\le \pi _{+j}\). Thus, \(\omega _{ij}=a_{ij}\pi _{+j}>0\), which implies \(E \subseteq E_1\). Similarly, we can prove for \(i=2\). \(\square \)

\(\mathrm{(i)}\Rightarrow \mathrm{(ii)}\):

Assume that A and B are g-compatible. From Definition 1, there exists a joint pdf \(\varvec{\pi }\) having support C such that \(\pi _{ij}=a_{ij}\pi _{+j}\) and \(\pi _{ij}=b_{ij}\pi _{i+}\) for all \((i,j) \in C\). Notice that , \(\varvec{\pi } T^{(1)}=\varvec{\pi }\) and \(\varvec{\pi } T^{(2)}=\varvec{\pi }\) are equivalent to \(\pi _{ij}=a_{ij}\pi _{+j}\) and \(\pi _{ij}=b_{ij}\pi _{i+}\) for all \((i,j) \in \Omega \). To complete the proof, we need to show \(\pi _{ij}=a_{ij}\pi _{+j}\) and \(\pi _{ij}=b_{ij}\pi _{i+}\) for all \((i,j) \notin C\). Suppose \((i,j) \notin C\), that is \((i,j)\in C_0^{(12)}(=C_0^{(21)})\). There are four possible cases to discuss: (a) \(a_{ij}>0\) and \(b_{ij}>0\), (b) \(a_{ij}>0\) and \(b_{ij}=0\), (c) \(a_{ij}=0\) and \(b_{ij}>0\) and (d) \(a_{ij}=0\) and \(b_{ij}=0\). We only prove case (a), the rest cases can be argued similarly. We claim all states \((\ell ,j)\)’s, \(1 \le \ell \le I\) and \(\ell \ne i\), are transient in \(T^{(12)}\), that is, they are in \(C_0^{(12)}\). Otherwise, (i, j) is a recurrent state in \(T^{(12)}\), which is a contradiction. This shows \(\pi _{\ell j}=0\) for \(1 \le \ell \le I\), which gives \(\pi _{+j}=0\). Similarly, we can show that \(\pi _{ik}=0\) for \(1 \le k \le J\), which gives \(\pi _{i+}=0\). Consequently, \(\pi _{ij}=a_{ij}\pi _{+j}\) and \(\pi _{ij}=b_{ij}\pi _{i+}\) for all \((i,j) \notin C\).

\(\mathrm{(ii)}\Rightarrow \mathrm{(i)}\):

Obviously.

\(\mathrm{(ii)}\Rightarrow \mathrm{(iii)}\):

Since \(\varvec{\pi } T^{(ij)}=\varvec{\pi } T^{(i)}T^{(j)}=\varvec{\pi } T^{(j)}=\varvec{\pi }\), hence, \(\varvec{\pi }\) is a stationary probability vector for \(T^{(ij)}\).

\(\mathrm{(iii)}\Rightarrow \mathrm{(ii)}\):

Since \(T^{(i)}\) is idempotent, we have \(\varvec{\pi }T^{(i)}=\varvec{\pi }T^{(ji)}T^{(i)}=\varvec{\pi }T^{(j)}T^{(i)}T^{(i)}\) \(= \varvec{\pi }T^{(j)}T^{(i)}=\varvec{\pi }T^{(ji)}=\varvec{\pi }\).

\(\mathrm{(iii)}\Rightarrow \mathrm{(iv)}\):

According to Theorem 3, \(\widetilde{\varvec{\pi }}_{k}^{(ij)} T^{(i)}=\widetilde{\varvec{\pi }}_{k}^{(ji)} \) for \(k=1,\ldots ,L\). By Lemma 4, it shows that the support of \(\widetilde{\varvec{\pi }}_k^{(ij)}\), which is \(C_k^{(ij)}\), is included in that of \(\widetilde{\varvec{\pi }}_k^{(ji)}\), which is \(C_k^{(ji)}\). This gives \(C_k^{(ij)}\subseteq C_k^{(ji)}\). Hence, \(C_k^{(12)}=C_k^{(21)} \equiv C_k\) for each \(1 \le k \le L\). Next, with a suitable order of states, we may assume that \(\varvec{\pi }=(\varvec{\pi }_0,\varvec{\pi }_1,\ldots ,\varvec{\pi }_L)\) is a common stationary probability vector for \(T^{(12)}\) and \(T^{(21)}\), where \(\varvec{\pi }_k\) corresponding to \(C_k\) is a sub-vector of \(\varvec{\pi }\). Note that \(\varvec{\pi }_{0}\) corresponding to the transient class is a zero vector. Therefore, \(\varvec{\pi }_k / \Vert \varvec{\pi }_k \Vert \) is the unique common stationary probability vector having support \(C_k\) for \(T_k^{(12)}\) and \(T_k^{(21)}\), where \(\Vert \varvec{\pi }_k \Vert \) is the \(L^{1}\)-norm for \(\varvec{\pi }_k\).

\(\mathrm{(iv)}\Rightarrow \mathrm{(iii)}\):

Suppose \({T}_k^{(12)}\) and \({T}_k^{(21)}\) share a unique positive stationary probability vector, say \(\varvec{\pi }_{k}\), with support \(C_k\), for \(k=1,\ldots ,L\). Let the weight vector \(\varvec{w}=(w_1,\ldots ,w_L)\), where \(w_k>0\) and \(\sum _{k=1}^{L} w_k=1\). Define \(\varvec{\pi } = (\varvec{0}, w_1\varvec{\pi }_1,\ldots ,w_L\varvec{\pi }_L)\), where \(\varvec{0}\) is an appropriate length zero vector corresponding to the transient class. Then, \(\varvec{\pi }\) is a common stationary probability vector with support \(C=\cup _{k=1}^L C_k\) for \(T^{(12)}\) and \(T^{(21)}\). Here, we may need to arrange the states in \(T^{(12)}\) and \(T^{(21)}\) so that their order is conformable with that in \(\varvec{\pi }\).

\(\mathrm{(iv)}\Rightarrow \mathrm{(v)}\):

Since \(T_k^{(12)}\) and \(T_k^{(21)}\) are irreducible and their diagonals have at least one positive entry, we have both \(T_k^{(12)}\) and \(T_k^{(21)}\) are primitive. In addition, \((T_k^{(12)})^\infty =\varvec{1}^{\prime }\varvec{\pi }_k=(T_k^{(21)})^{\infty }\), where \(\varvec{\pi }_k\) is the unique common stationary probability vector for \(T_k^{(12)}\) and \(T_k^{(21)}\).

\(\mathrm{(v)}\Rightarrow \mathrm{(iv)}\):

Since \((T_k^{(12)})^\infty =\varvec{1}^{\prime }\varvec{\pi }_k^{(12)}=\varvec{1}^{\prime }\varvec{\pi }_k^{(21)}=(T_k^{(21)})^\infty \), in which \(\varvec{\pi }_k^{(ij)}\) is the unique positive stationary probability vector for \(T^{(ij)}\). It follows that \(\varvec{\pi }_k^{(12)}=\varvec{\pi }_k^{(21)}\).

\(\square \)

We need only to prove C is the unique recurrent class in \(T^{(12)}\) and \(T^{(21)}\). Suppose on the contrary that \(T^{(12)}\) (and \(T^{(21)}\)) has multiple recurrent classes. Then, by Theorem 8 and the subsequent discussion, A and B would have multiple generalized joint pdfs with support C, which contradicts the assumption that A and B have a unique generalized joint pdf. Conversely, by Theorem 8, we have \(C_1^{(12)}=C_1^{(21)}=C\) and \((T_1^{(12)})^\infty =(T_1^{(21)})^\infty =\varvec{1}^{\prime } \varvec{\pi }\), where \(\varvec{\pi }\) is the unique stationary probability vector shared by \(T_1^{(12)}\) and \(T_1^{(21)}\). It follows that \((T^{(12)})^\infty =(T^{(21)})^\infty =\varvec{1}^{\prime } \widetilde{\varvec{\pi }}\), where \(\widetilde{\varvec{\pi }}\) is the augmented vector of \(\varvec{\pi }\) by appending zero probabilities to the components corresponding to the state in \(\Omega \backslash C\). Since \(\widetilde{\varvec{\pi }}\) is a common stationary probability vector for \(T^{(12)}\) and \(T^{(21)}\), and has support C, hence, \(\widetilde{\varvec{\pi }}\) is the unique generalized joint pdf of A and B by Theorem 8.

\(\square \)

First, we claim the existence of a stationary probability vector for \(\mathbb {T}_{\beta }\). Because \(\widetilde{\varvec{\pi }}^{(12)}_{[\varvec{\alpha }]}T^{(1)}=\widetilde{\varvec{\pi }}^{(21)}_{[\varvec{\alpha }]}\), \(\widetilde{\varvec{\pi }}^{(12)}_{[\varvec{\alpha }]}T^{(2)}=\widetilde{\varvec{\pi }}^{(12)}_{[\varvec{\alpha }]}\), \(\widetilde{\varvec{\pi }}^{(21)}_{[\varvec{\alpha }]}T^{(1)}=\widetilde{\varvec{\pi }}^{(21)}_{[\varvec{\alpha }]}\), and \(\widetilde{\varvec{\pi }}^{(21)}_{[\varvec{\alpha }]})T^{(2)}=\widetilde{\varvec{\pi }}^{(12)}_{[\varvec{\alpha }]}\), we have \((1-\beta )\widetilde{\varvec{\pi }}^{(12)}_{[\varvec{\alpha }]}+\beta \widetilde{\varvec{\pi }}^{(21)}_{[\varvec{\alpha }]}\mathbb {T}_{\beta }=(1-\beta )\widetilde{\varvec{\pi }}^{(12)}_{[\varvec{\alpha }]}+\beta \widetilde{\varvec{\pi }}^{(21)}_{[\varvec{\alpha }]}\).

Next, we explore all possible solutions. Suppose that \(\varvec{\pi }\) is a stationary probability vector for \(\mathbb {T}_{\beta }\). Because

and

we derive

due to \(0<\beta < 1\). Moreover, through Eq. (1), we have

and

That is, \(\varvec{\pi } T^{(2)}\) and \(\varvec{\pi } T^{(1)}\) are stationary probability vectors for \(T^{(12)}\) and \(T^{(21)}\), respectively. In addition, they are in pairs due to Eq. (1). By Theorem 4, Second Correspondence Theorem, we can present \(\varvec{\pi } T^{(2)}\) and \(\varvec{\pi } T^{(1)}\) as \(\widetilde{\varvec{\pi }}^{(12)}_{[\varvec{\alpha }]}\) and \(\widetilde{\varvec{\pi }}^{(21)}_{[\varvec{\alpha }]}\) for some \(\varvec{\alpha }\), respectively. Finally, we have \(\varvec{\pi } =\varvec{\pi } \mathbb {T}_{\beta }=\beta \widetilde{\varvec{\pi }}^{(21)}_{[\varvec{\alpha }]}+ (1-\beta )\widetilde{\varvec{\pi }}^{(12)}_{[\varvec{\alpha }]}\). \(\square \)

1. (i)

   To show \(\{C_k^*\}_{k=1}^L\) is the set of recurrent classes in \(\mathbb {T}_\beta \), we need to show \(C_k^*\) is irreducible and closed in \(\mathbb {T}_\beta \) for each k. Irreducibility. For any two states \(\varvec{u}\) and \(\varvec{v}\) in \(C_k^{(12)}\), there exists a sequence of states, say \(\{\varvec{v}_1,\ldots , \varvec{v}_m\}\), in \(C_k^{(12)}\) such that \(\varvec{u}{\mathop {\longrightarrow }\limits ^{12}}\varvec{v}_1 {\mathop {\longrightarrow }\limits ^{12}} \varvec{v}_2 {\mathop {\longrightarrow }\limits ^{12}} \cdots {\mathop {\longrightarrow }\limits ^{12}} \varvec{v}_m {\mathop {\longrightarrow }\limits ^{12}} \varvec{v}\), or, \(\varvec{u}{\mathop {\longrightarrow }\limits ^{1}}\varvec{u}_1 {\mathop {\longrightarrow }\limits ^{2}} \varvec{v}_1 {\mathop {\longrightarrow }\limits ^{1}}\varvec{u}_2 {\mathop {\longrightarrow }\limits ^{2}} \varvec{v}_2 {\longrightarrow } \cdots {\mathop {\longrightarrow }\limits ^{2}} \varvec{v}_m {\mathop {\longrightarrow }\limits ^{1}} \varvec{u}_{m+1} {\mathop {\longrightarrow }\limits ^{2}} \varvec{v}\), in which the states \(\varvec{u}_1,\ldots ,\varvec{u}_{m+1}\) are in \(C_k^{(21)}\) and \(t_{\varvec{u}\varvec{u}_1}^{(\beta )}=\beta a_{\varvec{u}_1}>0\), \(t_{\varvec{u}_1\varvec{v}_1}^{(\beta )}=(1-\beta ) b_{\varvec{v}_1}>0,\ldots , t_{\varvec{v}_m\varvec{u}_{m+1}}^{(\beta )}=\beta a_{\varvec{u}_{m+1}}>0\), \(t_{\varvec{u}_{m+1}\varvec{v}}^{(\beta )}=(1-\beta ) b_{\varvec{v}}>0\). This shows that \(\varvec{v}\) is accessible from \(\varvec{u}\) in \(C_k^*\). i.e., \(\varvec{u} {\mathop {\dashrightarrow }\limits ^{\mathbb {T_\beta }}} \varvec{v}\) in \(C_k^*\). Similarly, for any two states \(\varvec{u}\) and \(\varvec{v}\) in \(C_k^{(21)}\), we can show that \(\varvec{v}\) is is accessible from \(\varvec{u}\) in \(C_k^*\). i.e., \(\varvec{u} {\mathop {\dashrightarrow }\limits ^{\mathbb {T_\beta }}} \varvec{v}\) in \(C_k^*\). Now, suppose \(\varvec{u}\) is in \(C_k^{(12)}\) and \(\varvec{v}\) is in \(C_k^{(21)}\). By Theorem 1 (First Correspondence Theorem), there exists a state \(\varvec{w} \in C_k^{(12)}\) such that \(\varvec{w} {\mathop {\longrightarrow }\limits ^{1}} \varvec{v}\), which implies \(\varvec{w} {\mathop {\longrightarrow }\limits ^{\mathbb {T_\beta }}} \varvec{v}\). Since \(\varvec{u}\) and \(\varvec{w}\) are in \(C_k^{(12)}\), we have \(\varvec{u} {\mathop {\dashrightarrow }\limits ^{\mathbb {T_\beta }}} \varvec{w}\). It follows that \(\varvec{u} {\mathop {\dashrightarrow }\limits ^{\mathbb {T_\beta }}} \varvec{v}\). Hence, \(C_k^*\) is irreducible in \(\mathbb {T_\beta }\). Closed. To show \(C_k^*\) is closed, it suffices to show that \(t_{\varvec{u}\varvec{v}}^{(\beta )}=0\) for any \(\varvec{u}=(u_1,u_2) \in C_k^*\) and \(\varvec{v}=(v_1,v_2) \notin C_k^*\). Without loss of generality, we may assume \(\varvec{u} \in C_k^{(12)}\). It can be argued similarly for \(\varvec{u} \in C_k^{(21)}\). There are four possible cases to discuss.

   1. (a)

      \(a_{\varvec{v}}>0\) and \(b_{\varvec{v}}>0\) : Claim: \(u_1 \ne v_1\) and \(u_2 \ne v_2\). Suppose \(u_1 = v_1\). By Theorem 1, there exists \(\varvec{u}^* \in C_k^{(21)}\) such that \(\varvec{u}^*{\mathop {\longrightarrow }\limits ^{2}}\varvec{u}\). Since \(u_1=v_1\) and \(b_{\varvec{v}}>0\), we have \(\varvec{u}^*{\mathop {\longrightarrow }\limits ^{2}}\varvec{v}\). From \(a_{\varvec{v}}>0\), we have \(\varvec{v}{\mathop {\longrightarrow }\limits ^{1}}\varvec{v}\). Thus, \(\varvec{u}^*{\mathop {\longrightarrow }\limits ^{21}}\varvec{v}\), which implies that \(\varvec{v} \in C_k^{(21)} \subseteq C_k^*\). It is a contradiction. Thus, \(u_1 \ne v_1\). Suppose \(u_2 = v_2\). Since \(b_{\varvec{v}}>0\) , we then have \(\varvec{u}{\mathop {\longrightarrow }\limits ^{1}}\varvec{v}{\mathop {\longrightarrow }\limits ^{2}}\varvec{v}\). This implies that \(\varvec{u}{\mathop {\longrightarrow }\limits ^{12}}\varvec{v}\) and \(\varvec{v} \in C_k^{(12)} \subseteq C_k^*\). It contradicts to \(\varvec{v} \notin C_k^*\). Thus, \(u_2 \ne v_2\). It follows from \(u_1 \ne v_1\) and \(u_2 \ne v_2\) that \(t_{\varvec{u}\varvec{v}}^{(\beta )}=0\).

   2. (b)

      \(a_{\varvec{v}}>0\) and \(b_{\varvec{v}}=0\) : Claim: \(u_2 \ne v_2\). Suppose \(u_2=v_2\). By Theorem 1, there exists \(\varvec{u}^* \in C_k^{(21)}\) such that \(\varvec{u}^*{\mathop {\longrightarrow }\limits ^{2}}\varvec{u}\). Since \(u_2=v_2\) and \(a_{\varvec{v}}>0\), we have \(\varvec{u}^*{\mathop {\longrightarrow }\limits ^{2}}\varvec{u}{\mathop {\longrightarrow }\limits ^{1}}\varvec{v}\). Thus \(\varvec{u}^*{\mathop {\longrightarrow }\limits ^{21}}\varvec{v}\), which implies \(\varvec{v} \in C_k^{(21)} \subseteq C_k^*\). It contradicts to \(\varvec{v} \notin C_k^*\). Hence, \(u_2 \ne v_2\) and we are given \(t_{\varvec{u}\varvec{v}}^{(\beta )}=0\) whether or not \(u_1=v_1\).

   3. (c)

      \(a_{\varvec{v}}=0\) and \(b_{\varvec{v}}>0\) : Claim: \(u_1 \ne v_1\). Suppose \(u_1=v_1\). It gives \(\varvec{u}{\mathop {\longrightarrow }\limits ^{2}}\varvec{v}\). Since \(\varvec{u}\in C_k^{(12)}\), by Theorem 1, there exists \(\varvec{u}^* \in C_k^{(21)}\) such that \(\varvec{u}^*{\mathop {\longrightarrow }\limits ^{2}}\varvec{u}\). Hence \(\varvec{u}^*{\mathop {\longrightarrow }\limits ^{2}}\varvec{v}\). By Theorem 1 again, there exists \(\varvec{u}^{**} \in C_k^{(12)}\) such that \(\varvec{u}^{**}{\mathop {\longrightarrow }\limits ^{1}}\varvec{u}^*\). As a consequence, \(\varvec{u}^{**}{\mathop {\longrightarrow }\limits ^{12}}\varvec{v}\), which implies \(\varvec{v} \in C_k^{(12)} \subseteq C_k^*\). It contradicts to \(\varvec{v} \notin C_k^*\). Therefore \(u_1 \ne v_1\) and we are given \(t_{\varvec{u}\varvec{v}}^{(\beta )}=0\) whether or not \(u_2=v_2\).

   4. (d)

      \(a_{\varvec{v}}=b_{\varvec{v}}=0\) : It is obvious from the definition that \(t_{\varvec{u}\varvec{v}}^{(\beta )}=0\).

2. (ii)

   Since \(t_{\varvec{u}\varvec{u}}^{(\beta )}=\beta a_{\varvec{u}}+(1-\beta )b_{\varvec{u}}>0\) for \(\varvec{u} \in C_k^*\), \({\mathbb {T}_\beta }|_{C_k^*}\) is primitive. Hence, \(({\mathbb {T}_\beta }|_{C_k^*})^\infty \) exists and \(({\mathbb {T}_\beta }|_{C_k^*})^\infty =\varvec{1}^{\prime }\widehat{\varvec{\pi }}_{k,\beta }\), where \(\widehat{\varvec{\pi }}_{k,\beta }\) is the unique positive stationary probability vector for \({\mathbb {T}_\beta }|_{C_k^*}\). By Theorem 10, \(\widehat{\varvec{\pi }}_{k,\beta }=(1-\beta )\widehat{\varvec{\pi }}_k^{(12)}+\beta \widehat{\varvec{\pi }}_k^{(21)}\), where \(\widehat{\varvec{\pi }}_k^{(12)}\) and \(\widehat{\varvec{\pi }}_k^{(21)}\) are the augmented vector of \(\varvec{\pi }_k^{(12)}\) and \(\varvec{\pi }_k^{(21)}\), respectively, by appending zero to the components corresponding to the states in \(C_k^* \backslash C_k^{(12)}\) and \(C_k^* \backslash C_k^{(21)}\).

3. (iii)

   It follows from (ii) and Meyer (2000, p. 698). \(\square \)

\(\mathrm{(i)}\Rightarrow \mathrm{(ii)}\):

Since A and B are g-compatible, it implies from Theorem 8 that \(C_k^{(12)}=C_k^{(21)}=C_k=C_k^*\) and \(\widehat{\varvec{\pi }}_k^{(12)}=\widehat{\varvec{\pi }}_k^{(21)}=\varvec{\pi }_k^{(12)}=\varvec{\pi }_k^{(21)}=\varvec{\pi }_k\). Hence, \(\widehat{\varvec{\pi }}_{k,\beta }=(1-\beta )\widehat{\varvec{\pi }}_k^{(12)}+ \beta \widehat{\varvec{\pi }}_k^{(21)}=\varvec{\pi }_k\) is independent of \(0<\beta <1\).

\(\mathrm{(ii)}\Rightarrow \mathrm{(i)}\):

Suppose \((\mathbb {T}_\beta |_{C_k^*})^\infty =\varvec{1}^{\prime }\varvec{\pi }_k^*\), where \(\varvec{\pi }_k^*\) is independent of \(0<\beta <1\), \(k=1,\ldots ,L\). It follows from Theorem 11(ii) that \(\widehat{\varvec{\pi }}_k^{(12)}=\widehat{\varvec{\pi }}_k^{(21)}=\varvec{\pi }_k^*\), \(k=1,\dots ,L\). Hence, \(C_k^{(12)}=C_k^{(21)}=C_k=C_k^*\), which implies \(\varvec{\pi }_k^{(12)}=\varvec{\pi }_k^{(21)}=\varvec{\pi }_k\), \(k=1,\ldots ,L\). As a result, A and B are g-compatible by Theorem 8.

\(\square \)

For checking g-compatibility of A and B through Theorem 8(i) and (v), based on the canonical form and a lemma by Meyer (2000, p.695 and p. 698), we provide two computational procedures, for each \(k=1, \dots , L\), to find (a) \(C_k^{(12)}\) and \(C_k^{(21)}\), and (b) \((T_k^{(12)})^\infty \) and \((T_k^{(21)})^\infty \) in this Appendix 2. After converting A and B to \(T^{(1)}\) and \(T^{(2)}\), we will have two \(K \times K\) transition matrices \(T^{(12)}\) and \(T^{(21)}\), where \(K=I \times J\).

1. (a) 

   To determine whether a state is transient or recurrent and even to find the set of its affiliated recurrent class if this state is recurrent, we choose the row number (say \(i^{\text {th}}\) row, where \(i=1, 2,\ldots ,K\)) of a transition matrix T (either \(T^{(12)}\) or \(T^{(21)}\)) corresponding to this state. Now, generate a multiple Bernoulli observation (i.e., an observation from a multinomial distribution) from the \(i^{\text {th}}\) row of the matrix T. Let the column number of this first observation be denoted by \(i_1\). Then, generate the second multiple Bernoulli observation from the \({i_1}^{\text {st}}\) row of T and let the column number of this second observation be denoted by \(i_2\). Continue this process until we have \(M+n\) column observations. Discard the first M observations and keep only the last n observations. Let \(S_i\) be the set of the distinct column values among these n observations \(i_{M+1}, i_{M+2},\ldots ,i_{M+n}\). If \(S_i\) contains i, then \(S_i\) is a recurrent class. If \(S_i\) does not contain i, then i is a transient state. Although there are K \(S_i\)’s, we have only L distinct mutually exclusive sets of \(S_i\)’s, denoted by \(C_k, k=1,\dots ,L\). Note that the set of transient classes \(C_0\) is the set of i’s that its corresponding \(S_i\)’s do not contain i. Each of those \(S_i\)’s that do not contain i is the same as one of \(C_k\)’s, \(k=1,\dots ,L\). By letting \(T=T^{(12)}\), we have \(C_0^{(12)}\), \(C_k^{(12)}\), \(k=1, \ldots ,L\). Similarly, by letting \(T=T^{(21)}\), we have \(C_0^{(21)}\), \(C_k^{(21)}\), \(k=1,\ldots ,L\). Now, we can verify whether \(C_k^{(12)} = C_k^{(21)}\), for each \(k=1,\ldots ,L\).

2. (b) 

   From the \(k^{\text {th}}\) recurrent class \(C_k\), \(k=1, 2,\ldots ,L\), we can construct a sub-matrix \(T_k\) of T by keeping only those rows and columns of T that correspond to recurrent states in \(C_k\) . Now \(T_k\) is irreducible. Based on the second part of Theorem 8(v), our second step is to check \((T_k^{(12)})^\infty = (T_k^{(21)})^\infty \), for all k. If \(T_k\) has at least one positive diagonal element, by (A4) of Section 2.2, \(T_k\) is primitive. By (A2) of Section 2.2, the limiting matrix \(T_k^\infty \) of \(T_k^n\) exists and each of its rows is \(\varvec{\pi }_k\). In addition, the limiting state probability vector for \(T_k\) exists and is \(\varvec{\pi }_k\), independent of the initial state probability vector \(\varvec{v}_0\). A simple computational procedure to find \(\varvec{\pi }_k\) will be given next. One may use the identity vector \(\varvec{1}_k^i\), which has a 1 corresponding to the \(i^{\text {th}}\) state for \(T_k\), and zeros to the rest of states for \(T_k\), as an initial probability vector \(\varvec{v}_0\). To find the \(m^{\text {th}}\) step probability vector \(\varvec{v}_m\), one can use a traditional matrix operation (e.g., eigenvalue-eigenvector) method or a sampling method. To use a sampling method, we generate a multiple Bernoulli observation from the \(i^{\text {th}}\) row of the matrix \(T_k\). Let the column number of this first observation as \(i_1\). Then, generate the second multiple Bernoulli observation from the \({i_1}^{\text {st}}\) row of \(T_k\) and let the column number of this second observation as \(i_2\). Continue this process until we have m column observations. Keep only the \({i_m}^{\text {th}}\) observation. Repeat this process n times (sample size n can be determined based on the desired multinomial margin of error), so we have n \({i_m}^{\text {th}}\) column observations. Let the \(j^{\text {th}}\) entry of \(\varvec{\pi }_k\) be the proportion of these n observations that reside in the \(j^{\text {th}}\) column. By letting \(T_k=T_k^{(12)}\), we have \(\varvec{\pi }_k=\varvec{\pi }_k^{(12)}\). Similarly, letting \(T_k=T_k^{(21)}\), we then have \(\varvec{\pi }_k=\varvec{\pi }_k^{(21)}\). Checking the second part of Theorem 8(v) is the same as checking \(\varvec{\pi }_k^{(12)}=\varvec{\pi }_k^{(21)}\), for all \(k=1,\dots ,L\).

• Section 2 :

  1. 1

     \(\Omega = \Omega _1\times \Omega _2\) : the state space, where \(\Omega _1=\{1,\ldots ,I\}\) and \(\Omega _2=\{1,\ldots ,J\}\) are the possible values of discrete random variables \(X_1\) and \(X_2\), respectively.

  2. 2

     \(A=[a_{ij}]\) : the conditional probability matrix of \(X_1 \mid X_2\), where \(a_{ij}=\Pr (X_1=i \mid X_2=j)\).

  3. 3

     \(B=[b_{ij}]\) : the conditional probability matrix of \(X_2 \mid X_1\), where \(b_{ij}=\Pr (X_2=i \mid X_1=j)\).

  4. 4

     \(N^{(1)}=\{(i,j): a_{ij}>0\}\) and \(N^{(2)}=\{(i,j): b_{ij}>0\}\): the supports of A and B, respectively.

  5. 5

     \(\varvec{\pi }=(\pi _{11},\ldots ,\pi _{I1},\pi _{12},\ldots ,\pi _{I2},\ldots ,\pi _{1J},\ldots ,\pi _{IJ})\) : a joint probability vector of \(X_1\) and \(X_2\).

  6. 6

     \(\varvec{u}=(u_1, u_2)\) : a state in \(\Omega \).

  7. 7

     \(T^{(1)}\) and \(T^{(2)}\): the transition matrices based on A and B, respectively.

  8. 8

     \(T^{(ij)}\equiv T^{(i)}T^{(j)}\) : the transition matrix of the Markov chain corresponding to the systematic-scan Gibbs sampler with scan order (i, j), where \(i \ne j\).

  9. 9

     \(\mathcal {U}=\{\varvec{\pi }: \pi _{\varvec{u}}\ge 0,\varvec{u}\in \Omega ,\pi _{++}=1\}\) : the set of all putative joint pdfs of A and B, where \(\pi _{\varvec{u}}\) is the \(\varvec{u}^{th}\) component of \(\varvec{\pi }\).

     • \(\mathcal {D}^{(1)}=\{\varvec{\pi }\in \mathcal {U}: \pi _{\varvec{u}}>0 \text { for } \varvec{u}\in N^{(1)}\}\) : the set of all putative joint pdfs of A and B with supports \(N^{(1)}\).

     • \(\mathcal {D}^{(2)}=\{\varvec{\pi }\in \mathcal {U}: \pi _{\varvec{u}}>0 \text { for } \varvec{u}\in N^{(2)}\}\) : the set of all putative joint pdfs of A and B with supports \(N^{(2)}\).

  10. 10

      \(\mathcal {C}^{(1)}=\{\varvec{\pi }\in \mathcal {D}^{(1)}: \pi _{\varvec{u}}/\pi _{+u_2}=a_{\varvec{u}}\}\) and \(\mathcal {C}^{(2)}=\{\varvec{\pi }\in \mathcal {D}^{(2)}: \pi _{\varvec{u}}/\pi _{u_1+}=b_{\varvec{u}}\}\) : the sets of all joint pdfs having conditional probability matrices A and B, respectively.

  11. 11

      \(\varvec{u}{\mathop {\longrightarrow }\limits ^{i}}\varvec{v}\) : state \(\varvec{v}\) is reachable from state \(\varvec{u}\) via \(T^{(i)}\) in one step, where \(i=1,2\).

  12. 12

      \(\varvec{u}{\mathop {\longrightarrow }\limits ^{ij}}\varvec{v}\) : state \(\varvec{v}\) is reachable from state \(\varvec{u}\) via \(T^{(ij)}\) in one step, where \(i \ne j\).

  13. 13

      \(\varvec{u}{\mathop {\dashrightarrow }\limits ^{ij}}\varvec{v}\) : state \(\varvec{v}\) is reachable from state \(\varvec{u}\) via \(T^{(ij)}\) in some steps, where \(i \ne j\).

  14. 14

      \(C_0^{(ij)}\) : the transient class of \(T^{(ij)}\), where \(i \ne j\).

  15. 15

      \(C_k^{(ij)}\) : the \(k^{\text {th}}\) recurrent class of \(T^{(ij)}\), where \(i \ne j\).

  16. 16

      \((C^{(ij)}_{k})_{X_t}=\{u_t: \varvec{u} \in C^{(ij)}_{k}\}\) : the \(X_t\)-projection of \(C^{(ij)}_{k}\), where \(t=1,2\) and \(i \ne j\).

  17. 17

      \(T_k^{(ij)}\) : the sub-matrix of \(T^{(ij)}\) restricted to the recurrent class \(C^{(ij)}_{k}\), where \(i \ne j\).

  18. 18

      \(\varvec{\pi }_k^{(ij)}\) : the unique positive stationary probability vector for \(T_k^{(ij)}\), where \(i \ne j\).

  19. 19

      \(\widetilde{\varvec{\pi }}^{(ij)}_k\) : the \(k^{\text {th}}\) elementary stationary probability vector for \(T^{(ij)}\), obtained from \(\varvec{\pi }^{(ij)}_k\) by appending zero probabilities to the components corresponding to states not in \(C^{(ij)}_{k}\), where \(i \ne j\).

  20. 20

      \(\mathcal {V}^{(ij)}=\{\widetilde{\varvec{\pi }}_{[\varvec{\alpha }]}^{(ij)}: \widetilde{\varvec{\pi }}_{[\varvec{\alpha }]}^{(ij)}=\sum _{k=1}^L\alpha _k\widetilde{\varvec{\pi }}^{(ij)}_k, \varvec{\alpha }=(\alpha _1,\ldots ,\alpha _L), \alpha _k\ge 0,\alpha _+=1\}\) : the set of all stationary probability vectors for \(T^{(ij)}\), where \(i \ne j\).

• Section 3 :

  1. 1

     \(\mathbb {T}_{\beta } \equiv \beta T^{(1)} +(1-\beta )T^{(2)}\) : the transition matrix of the Markov chain associated with the random-scan Gibbs sampler with selection probability \(0< \beta <1\).

  2. 2

     \(C_k^*\) : the \(k^{\text {th}}\) recurrent class of \(\mathbb {T}_{\beta }\).

  3. 3

     \(\mathbb {T}_\beta |_{C_k^*}\) : the sub-transition matrix of \(\mathbb {T}_\beta \) restricted to \(C_k^*\).

  4. 4

     \(\widehat{\varvec{\pi }}_k^{(ij)}\) : the augmented vector of \(\varvec{\pi }_k^{(ij)}\) obtained by appending zero probabilities to the components corresponding to the states in \(C_k^*\) but not in \(C_k^{(ij)}\), where \(i \ne j\).

  5. 5

     \(\widehat{\varvec{\pi }}_{k,\beta }=(1-\beta )\widehat{\varvec{\pi }}_k^{(12)}+ \beta \widehat{\varvec{\pi }}_k^{(21)}\) : the unique positive stationary probability vector for \(\mathbb {T}_\beta |_{C_k^*}\).

  6. 6

     \(\varvec{\pi }_k^*\) : the unique positive stationary probability vector for \(\mathbb {T}_\beta |_{C_k^*}\), independent of \(\beta \) for \(0< \beta <1\).