Appendix 1
The proof of Proposition 1.
Proof
Without loss of generality, we consider the case where \(\eta =1\). For the claim (i), note that \(B_1=A_{3,max}+A_{2,max}\), and
$$\begin{aligned}&p_{T_{1,0}^f}(k)\equiv P(T_{1,0}^f=k)\\&\quad =\left\{ \begin{array}{lll} \beta _1Q_1^{k-1}R_1 &{} \text{ if } k\ge A_{3,max}+A_{2,max}-1 \\ 0 &{} \text{ otherwise } \\ \end{array} \right. \end{aligned}$$
where \(Q_1^{A_{3,max}+A_{2,max}-1}R_1=f_3^{A_{3,max}}f_2^{A_{2,max}}\beta _1^T\). We have
$$\begin{aligned} \psi _{1,0}^f(z)=\sum _{k=0}^{\infty }z^kp_{T_{1,0}^f}(k)=(f_3z)^{A_{3,max}}(f_2z)^{A_{2,max}})\\&\quad \beta _1(I_1-zQ_1)^{-1}\beta _1^T \end{aligned}$$
and
$$\begin{aligned} E[T_{1,0}^f]&= {} (\psi _{1,0}^f(z))'_{z=1}\\&=f_3^{A_{3,max}}f_2^{A_{2,max}}\beta _1(I_1-Q_1)^{-2}k)\\&\quad ((A_{3,max}+A_{2,max})(I_1-Q_1)+Q_1)\beta _1^T. \end{aligned}$$
The direct calculation yields
$$\begin{aligned} (I_1-Q_1)^{-1}=\left[ \begin{array}{ccccccccccc} \frac{1}{f_3^{A_{3,max}}f_2^{A_{2,max}}} &{} \frac{1}{f_3^{A_{3,max}-1}f_2^{A_{2,max}}} &{} \cdots &{} \frac{1}{f_2^{A_{2,max}}} &{} \cdots &{} \frac{1}{f_2} \\ \frac{1-f_3^{A_{3,max}-1}f_2^{A_{2,max}}}{f_3^{A_{3,max}}f_2^{A_{2,max}}} &{} \frac{1}{f_3^{A_{3,max}-1}f_2^{A_{2,max}}} &{} \cdots &{} \frac{1}{f_2^{A_{2,max}}} &{} \cdots &{} \frac{1}{f_2} \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \ddots &{} \vdots \\ \frac{1-f_2^{A_{2,max}}}{f_3^{A_{3,max}}f_2^{A_{2,max}}} &{} \frac{1-f_2^{A_{2,max}}}{f_3^{A_{3,max}-1}f_2^{A_{2,max}}} &{} \cdots &{} \frac{1}{f_2^{A_{2,max}}} &{} \cdots &{} \frac{1}{f_2} \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \ddots &{} \vdots \\ \frac{1-f_2}{f_3^{A_{3,max}}f_2^{A_{2,max}}} &{} \frac{1-f_2}{f_3^{A_{3,max}-1}f_2^{A_{2,max}}} &{} \cdots &{} \frac{1}{f_2^{A_{2,max}}} &{} \cdots &{} \frac{1}{f_2} \\ \end{array} \right] . \end{aligned}$$
Substituting it into \(E[T_{1,0}^f]\), we obtain the claim (i). Next, let \(p_{T_{1,B_1}^f}(k)=P(T_{1,B_1}^f=k, I_{\{T_{1,B_1}^f<T_{1,0}^f\}})\). It follows from the transition matrix \(P_{1}^f\) of the Markov chain \(Y_1^f\) that
$$\begin{aligned} p_{T_{1,B_1}^f}(k)=\left\{ \begin{array}{lll} f_3^{k-1}(1-f_3) &{} \text{ if } 1\le k\le A_{3,max} \\ f_3^{A_{3,max}}f_2^{k-A_{3,max}-1}(1-f_2) &{} \text{ if } A_{3,max}< k\le A_{3,max}+A_{2,max} \\ 0 &{} \text{ otherwise }. \\ \end{array} \right. \end{aligned}$$
Using this result, the claim (ii) can be easily derived. Finally, we can establish the assert (iii) according to the following two facts: (1) the PGF of the sojourn time at the state \(B_1\) is \(\phi _{1,su}(z)+\phi _{1,co}(z)\), and the PGF of the sojourn time at the other states is z; (2) \(P_{1,B_1}^{f}\) is the rate that the freezing countdown process \(Y_1^f\) visits the state \(B_1\) before reach the absorption state 0. This completes the proof. \(\square \)
Appendix 2
The sub-matrices of the transition probability matrix \(\mathbf{P}_{\eta }\)
$$\begin{aligned} {\bar{\mathbf{A}}}_{0}^{\eta }&= {} \left[
\begin{array}{lllll} g_{\eta }r_{\eta ,0}+(1-g_{\eta })q_{\eta ,0}
&{} 0 &{} \cdots &{} 0 \\ \alpha ^{(1)}_{\eta ,0} &{} 0 &{} \cdots
&{} 0 \\ \alpha ^{(2)}_{\eta ,0} &{} 0 &{} \cdots &{} 0 \\ \vdots
&{} \vdots &{} \ddots &{} \vdots \\ \alpha ^{(\omega _{\eta
,0}-1)}_{\eta ,0} &{} 0 &{} \cdots &{} 0 \end{array} \right]
_{\omega _{\eta ,0}\times \omega _{\eta ,0}} \quad \text{ and
} \rm{ for } k\ge 1 \\ {\bar{\mathbf{A}}}_{k}^{\eta }&= {} \left[
\begin{array}{cccccccccccc} g_{\eta }r_{k}+\frac{(1-g_{\eta
})q_{\eta ,k}}{\omega _{\eta ,0}} &{} \frac{(1-g_{\eta })q_{\eta
,k}}{\omega _{\eta ,0}} &{} \cdots &{} \frac{(1-g_{\eta })q_{\eta
,k}}{\omega _{\eta ,0}} &{} 0 &{} \cdots &{} 0 \\ \alpha
^{(1)}_{\eta ,k} &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{} 0 \\
\alpha ^{(2)}_{\eta ,k} &{} 0 &{} \cdots &{} 0 &{} 0 &{} \cdots &{}
0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{} \ddots
&{} \vdots \\ \alpha ^{(\omega _{\eta ,0}-1)}_{\eta ,k} &{} 0 &{}
\cdots &{} 0 &{} 0 &{} \cdots &{} 0 \\ \end{array} \right] _{\omega
_{\eta ,0} \times \omega _{\eta ,k}.}\\ {\bar{\mathbf{B}}}_{0}^{\eta
}&= {} \left[ \begin{array}{llllll} {\bar{\mathbf{B}}}_{00} \\
{\bar{\mathbf{B}}}_{10} \\ \vdots \\ {\bar{\mathbf{B}}}_{m_{\eta }0}
\\ \end{array} \right] \quad \text{ and } \\ \mathbf{B}_{0}^{\eta }&= {} \left[ \begin{array}{llll} \mathbf{B}_{00} &{} 0 &{} \cdots &{}
0 \\ \mathbf{B}_{10} &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{}
\ddots &{} \vdots \\ \mathbf{B}_{m_{\eta }0} &{} 0 &{} \cdots &{} 0
\end{array} \right] _\mathrm{(m_{\eta }+1)\times (m_{\eta }+1)}
\end{aligned}
$$
where for \(i=0,1,\ldots ,m_{\eta },\)
$$\begin{aligned} {\bar{\mathbf{B}}}_{i0}&= {} \mathbf{B}_{i0}=\left[ \begin{array}{ccccccccc} \frac{g_{\eta }d_{\eta ,0}^{(1)}}{\omega _{\eta ,0}} &{} \frac{g_{\eta }d_{\eta ,0}^{(1)}}{\omega _{\eta ,0}} &{} \cdots &{} \frac{g_{\eta }d_{\eta ,0}^{(1)}}{\omega _{\eta ,0}} \\ 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 \end{array} \right] _{\omega _{\eta ,i}\times \omega _{\eta ,0}.}\\ \mathbf{B}_{k}^{\eta }&= {} \left[ \begin{array}{lcccccccccccc} \mathbf{B}^{(k)}_{00} &{} \mathbf{B}^{(k)}_{01} &{} 0 &{} \cdots &{} 0 &{} 0 \\ \mathbf{B}^{(k)}_{10} &{} \mathbf{B}^{(k)}_{11} &{} \mathbf{B}^{(k)}_{12} &{} \cdots &{} 0 &{} 0 \\ \mathbf{B}^{(k)}_{20} &{} 0 &{} \mathbf{B}^{(k)}_{22} &{} \cdots &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ \mathbf{B}^{(k)}_{m_{\eta }-10} &{} 0 &{} 0 &{} \cdots &{} \mathbf{B}^{(k)}_{m_{\eta }-1m_{\eta }-1} &{} \mathbf{B}^{(k)}_{m_{\eta }-1m_{\eta }} \\ \mathbf{B}^{(k)}_{m_{\eta }0} &{} 0 &{} 0 &{} \dots &{} 0 &{} \mathbf{B}^{(k)}_{m_{\eta }m_{\eta }} \end{array} \right] \end{aligned}$$
where
$$\begin{aligned} \mathbf{B}^{(k)}_{00}&= {} \left[ \begin{array}{ccccccccccccc} \frac{g_{\eta }d_{\eta ,k}^{(1)}}{\omega _{\eta ,0}} &{} \frac{g_{\eta }d_{\eta ,k}^{(1)}}{\omega _{\eta ,0}} &{} \cdots &{} \frac{g_{\eta }d_{\eta ,k}^{(1)}}{\omega _{\eta ,0}} &{} \frac{g_{\eta }d_{\eta ,k}^{(1)}}{\omega _{\eta ,0}} \\ d^*_{\eta ,k-1} &{} 0 &{} \cdots &{} 0 &{} 0 \\ 0 &{} d^*_{\eta ,k-1} &{} \cdots &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} d_{\eta ,k-1} &{} 0 \end{array} \right] _{\omega _{\eta ,0} \times \omega _{\eta ,0},}\\ {\mathbf{B}}^{(k)}_{i0}&= {} \left[ \begin{array}{cccccccccc} \frac{g_{\eta }d_{\eta ,k}^{(1)}}{\omega _{\eta ,0}} &{} \frac{g_{\eta }d_{\eta ,k}^{(1)}}{\omega _{\eta ,0}} &{} \cdots &{} \frac{g_{\eta }d_{\eta ,k}^{(1)}}{\omega _{\eta ,0}} \\ 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 \end{array} \right] _{\omega _{\eta ,i} \times \omega _{\eta ,0},} \\ \mathbf{B}^{(k)}_{ii}&= {} \left[ \begin{array}{cccccccccccccccc} 0 &{} 0 &{} \cdots &{} 0 &{} 0 \\ d^*_{\eta ,k-1} &{} 0 &{} \cdots &{} 0 &{} 0 \\ 0 &{} d^*_{\eta ,k-1} &{} \cdots &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} d_{\eta ,k-1} &{} 0 \end{array} \right] _{\omega _{\eta ,i} \times \omega _{\eta ,i},} \\ \mathbf{B}^{(k)}_{ii+1}&= {} \left[ \begin{array}{ccccccccc} \frac{(1-g_{\eta })d^{co}_{\eta ,k-1}}{\omega _{\eta ,i+1}} &{} \frac{(1-g_{\eta })d^{co}_{\eta ,k-1}}{\omega _{\eta ,i+1}} &{} \cdots &{} \frac{(1-g_{\eta })d^{co}_{\eta ,k-1}}{\omega _{\eta ,i+1}} \\ 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 \end{array} \right] _{\omega _{\eta ,i} \times \omega _{\eta ,i+1},}\\ \mathbf{B}^{(k)}_{m_{\eta }m_{\eta }}&= {} \left[ \begin{array}{ccccccccccccccccc} \frac{(1-g_{\eta })d^{co}_{\eta ,k-1}}{\omega _{\eta ,m_{\eta }}} &{} \frac{(1-g_{\eta })d^{co}_{\eta ,k-1}}{\omega _{\eta ,m_{\eta }}} &{} \cdots &{} \frac{(1-g_{\eta })d^{co}_{\eta ,k-1}}{\omega _{\eta ,m_{\eta }}} &{} \frac{(1-g_{\eta })d^{co}_{\eta ,k-1}}{\omega _{\eta ,m_{\eta }}} \\ d^*_{\eta ,k-1} &{} 0 &{} \cdots &{} 0 &{} 0 \\ 0 &{} d^*_{\eta ,k-1} &{} \cdots &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} d_{\eta ,k-1} &{} 0 \end{array} \right] _{\omega _{\eta ,m_{\eta }} \times \omega _{\eta ,m_{\eta }}.} \end{aligned}$$
Furthermore,
$$\begin{aligned} {\bar{\mathbf{K}}}_{0}^{\eta }&= {} \left[ \begin{array}{cccccccccccccccc} {\bar{\mathbf{K}}}_{00} \\ \mathbf{0} \\ \vdots \\ \mathbf{0} \\ \end{array} \right] _\mathrm{(m_{\eta }+1) \times 1} \quad \text{ where } \\ {\bar{\mathbf{K}}}_{00}&= {} \left[ \begin{array}{ccccccccccccccc} \frac{g_{\eta }d_{\eta ,0}^{(N_{\eta })}}{\omega _{\eta ,0}} &{} \frac{g_{\eta }d_{\eta ,0}^{(N_{\eta })}}{\omega _{\eta ,0}} &{} \cdots &{} \frac{g_{\eta }d_{\eta ,0}^{(N_{\eta })}}{\omega _{\eta ,0}} \\ 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{}\vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 \end{array} \right] _{\omega _{\eta ,0}\times \omega _{\eta ,0}.} \end{aligned}$$
For \(k=0,1 \cdots , N_{\eta }-2\),
$$\begin{aligned} \mathbf{K}_{k}^{\eta }&= {} \left[ \begin{array}{cccccccccccccc} \mathbf{K}^{(k)}_{00} &{} \mathbf{0} &{} \cdots &{} \mathbf{0} \\ \mathbf{0} &{} \mathbf{0} &{} \cdots &{} \mathbf{0} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \mathbf{0} &{} \mathbf{0}&{} \cdots &{} \mathbf{0} \end{array} \right] _{(m_{\eta }+1) \times (m_{\eta }+1),} \quad \text{ where }\\ \mathbf{K}^{(k)}_{00}&= {} \left[ \begin{array}{ccccccccccccc} \frac{g_{\eta }d_{\eta ,k}^{(N_{\eta })}}{\omega _{\eta ,0}} &{} \frac{g_{\eta }d_{\eta ,k}^{(N_{\eta })}}{\omega _{\eta ,0}} &{} \cdots &{} \frac{g_{\eta }d_{\eta ,k}^{(N_{\eta })}}{\omega _{\eta ,0}} \\ 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{}\vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 \end{array} \right] _{\omega _{\eta ,0}\times \omega _{\eta ,0}.} \end{aligned}$$
and
$$\begin{aligned} \mathbf{K}_{(N-1)}^{\eta }&= {} \left[ \begin{array}{cccccccccccccccccc} \mathbf{K}^{(N-1)}_{00} &{} \mathbf{0} &{} \cdots &{} \mathbf{0} \\ \mathbf{K}^{(N-1)}_{10} &{} \mathbf{0} &{} \cdots &{} \mathbf{0} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \mathbf{K}^{(N-1)}_{m_{\eta }0} &{} \mathbf{0} &{} \cdots &{} \mathbf{0} \end{array} \right] _\mathrm{(m_{\eta }+1)\times (m_{\eta }+1)} \quad \text{ where }\\ \mathbf{K}^{(N_{\eta }-1)}_{i0}&= {} \left[ \begin{array}{cccccccccccccccc} \frac{g_{\eta }d_{\eta ,0}^{(1)}}{\omega _{\eta ,0}} &{} \frac{g_{\eta }d_{\eta ,0}^{(1)}}{\omega _{\eta ,0}} &{} \cdots &{} \frac{g_{\eta }d_{\eta ,0}^{(1)}}{\omega _{\eta ,0}}\\ 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 \end{array} \right] _{\omega _{\eta ,i} \times \omega _{\eta ,0}.} \end{aligned}$$
and
$$\begin{aligned} \mathbf{K}^{(N_{\eta }-1)}_{00}&= {} \left[ \begin{array}{cccccccccccccccc} \frac{g_{\eta }d_{\eta ,N_{\eta }-1}^{(N_{\eta })} }{\omega _{\eta ,0}} &{} \frac{g_{\eta }d_{\eta ,N_{\eta }-1}^{(N_{\eta })} }{\omega _{\eta ,0}} &{} \cdots &{} \frac{g_{\eta }d_{\eta ,N_{\eta }-1}^{(N_{\eta })} }{\omega _{\eta ,0}} \\ 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{}\ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 \end{array} \right] _{\omega _{\eta ,0} \times \omega _{\eta ,0}.} \end{aligned}$$
Finally, for \(k=N_{\eta },N_{\eta }+1,\ldots \)
$$\begin{aligned} \mathbf{K}_{k}^{\eta }&= {} \left[ \begin{array}{cccccccccccccc} \mathbf{K}^{(k)}_{00} &{} \mathbf{K}^{(k)}_{01} &{} \mathbf{0} &{} \cdots &{} \mathbf{0} &{} \mathbf{0} \\ \mathbf{K}^{(k)}_{10} &{} \mathbf{K}^{(k)}_{11} &{} \mathbf{K}^{(k)}_{12} &{} \cdots &{} \mathbf{0} &{} \mathbf{0} \\ \mathbf{K}^{(k)}_{20} &{} \mathbf{0} &{} \mathbf{K}^{(k)}_{22} &{} \cdots &{} \mathbf{0} &{} \mathbf{0} \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ \mathbf{K}^{(k)}_{m_{\eta }-10} &{} \mathbf{0} &{} \mathbf{0} &{} \cdots &{} \mathbf{K}^{(k)}_{m_{\eta }-1m_{\eta }-1} &{} \mathbf{K}^{(k)}_{m_{\eta }-1m_{\eta }} \\ \mathbf{K}^{(k)}_{m_{\eta }0} &{} \mathbf{0} &{} \mathbf{0} &{} \cdots &{} \mathbf{0} &{} \mathbf{K}^{(k)}_{m_{\eta }m_{\eta }} \end{array} \right] \end{aligned}$$
where
$$\begin{aligned} \mathbf{K}^{(k)}_{00}&= {} \left[ \begin{array}{ccccccccccc} \frac{g_{\eta }d_{\eta ,k}^{(N_{\eta })}}{\omega _{\eta ,0}} &{} \frac{g_{\eta }d_{\eta ,k}^{(N_{\eta })}}{\omega _{\eta ,0}} &{} \cdots &{} \frac{g_{\eta }d_{\eta ,k}^{(N_{\eta })}}{\omega _{\eta ,0}} &{} \frac{g_{\eta }d_{\eta ,k}^{(N_{\eta })}}{\omega _{\eta ,0}} \\ d^*_{\eta ,k-N_{\eta }} &{} 0 &{} \cdots &{} 0 &{} 0 \\ 0 &{} d^*_{\eta ,k-N_{\eta }} &{} \cdots &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} d_{\eta ,k-N_{\eta }} &{} 0 \end{array} \right] _{\omega _{\eta ,0} \times \omega _{\eta ,0}} \end{aligned}$$
and
$$\begin{aligned} \mathbf{K}^{(k)}_{i0}&= {} \left[ \begin{array}{ccccccccccc} \frac{g_{\eta }d_{\eta ,k-N_{\eta }+1}^{(N_{\eta })}}{\omega _{\eta ,0}} &{} \frac{g_{\eta }d_{\eta ,k-N_{\eta }+1}^{(N_{\eta })}}{\omega _{\eta ,0}} &{} \cdots &{} \frac{g_{\eta }d_{\eta ,k-N_{\eta }+1}^{(N_{\eta })}}{\omega _{\eta ,0}} &{} \frac{g_{\eta }d_{\eta ,k-N_{\eta }+1}^{(N_{\eta })}}{\omega _{\eta ,0}} \\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 \\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 \end{array} \right] _{\omega _{\eta ,i} \times \omega _{\eta ,0},}\\ \mathbf{K}^{(k)}_{ii}&= {} \left[ \begin{array}{ccccccccccc} 0 &{} 0 &{} \cdots &{} 0 &{} 0 \\ d^*_{\eta ,k-N_{\eta }} &{} 0 &{} \cdots &{} 0 &{} 0 \\ 0 &{} d^*_{\eta ,k-N_{\eta }} &{} \cdots &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} d_{\eta ,k-N_{\eta }} &{} 0 \end{array} \right] _{\omega _{\eta ,i} \times \omega _{\eta ,i},}\\ \mathbf{K}^{(k)}_{ii+1}&= {} \left[ \begin{array}{ccccccccc} \frac{(1-g_{\eta })d^{co}_{\eta ,k-N_{\eta }}}{\omega _{\eta ,i+1}} &{} \frac{(1-g_{\eta })d^{co}_{\eta ,k-N_{\eta }}}{\omega _{\eta ,i+1}} &{} \cdots &{} \frac{(1-g_{\eta })d^{co}_{\eta ,k-N_{\eta }}}{\omega _{\eta ,i+1}} \\ 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 \end{array} \right] _{\omega _{\eta ,i} \times \omega _{\eta ,i+1},}\\ \mathbf{K}^{(k)}_{m_{\eta }m_{\eta }}&= {} \left[ \begin{array}{ccccccccc} \frac{(1-g_{\eta })d^{co}_{k-N_{\eta }}}{\omega _{\eta ,m_{\eta }}} &{} \frac{(1-g_{\eta })d^{co}_{k-N_{\eta }}}{\omega _{\eta ,m_{\eta }}} &{} \cdots &{} \frac{(1-g_{\eta })c_{k-N_{\eta }}}{\omega _{\eta ,m_{\eta }}} &{} \frac{(1-g_{\eta })d^{co}_{k-N_{\eta }}}{\omega _{\eta ,m_{\eta }}} \\ d^*_{\eta ,k-N_{\eta }} &{} 0 &{} \cdots &{} 0 &{} 0 \\ 0 &{} d^*_{\eta ,k-N_{\eta }} &{} \cdots &{} 0 &{} 0 &{} \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} d_{\eta ,k-N_{\eta }} &{} 0 \end{array} \right] _{\omega _{\eta ,m_{\eta }} \times \omega _{\eta ,m_{\eta }}.} \end{aligned}$$
Appendix 3
(1) The definition of the sub-matrices of\(K_{\eta }(z)\)
$$\begin{aligned} K_{00}(z)&= {} \left[ \begin{array}{cccccccc} \frac{g_{\eta }}{\omega _{\eta ,0}}d^{(N_{\eta })}_{\eta }(z) &{} \frac{g_{\eta }}{\omega _{\eta ,0}}d^{(N_{\eta })}_{\eta }(z) &{} \cdots &{} \frac{g_{\eta }}{\omega _{\eta ,0}}d^{(N_{\eta })}_{\eta }(z) &{} \frac{g_{\eta }}{\omega _{\eta ,0}}d^{(N_{\eta })}_{\eta }(z) \\ z^{N_{\eta }}d_{\eta }^*(z) &{} 0 &{} \cdots &{} 0 &{} 0 \\ 0 &{} z^{N_{\eta }}d_{\eta }^*(z) &{} \cdots &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} z^{N_{\eta }}d_{\eta }(z) &{} 0 \end{array} \right] _{\omega _{\eta ,0}\times \omega _{\eta ,0}} \end{aligned}$$
For \(i=1,\ldots , m_{\eta },\)
$$\begin{aligned} K_{i0}(z)&= {} \left[ \begin{array}{cccccccc} \frac{g_{\eta }}{\omega _{\eta ,0}}z^{N_{\eta }-1}d_{\eta }^{(1)}(z) &{} \frac{g_{\eta }}{\omega _{\eta ,0}}z^{N_{\eta }-1}d_{\eta }^{(1)}(z) &{} \cdots &{} \frac{g_{\eta }}{\omega _{\eta ,0}}z^{N_{\eta }-1}d_{\eta }^{(1)}(z) \\ 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 \end{array} \right] _{\omega _{\eta ,i}\times \omega _{\eta ,0}} \end{aligned}$$
and for \(i=1,\ldots , m_{\eta }-1,\)
$$\begin{aligned} K_{ii+1}(z)&= {} \left[ \begin{array}{cccccccc} \frac{g_{\eta }}{\omega _{\eta ,i+1}}z^{N_{\eta }}d_{\eta }^{co}(z) &{} \frac{g_{\eta }}{\omega _{\eta ,i+1}}z^{N_{\eta }}d_{\eta }^{co}(z) &{} \cdots &{} \frac{g_{\eta }}{\omega _{\eta ,i+1}}z^{N_{\eta }}d_{\eta }^{co}(z) \\ 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 \end{array} \right] _{\omega _{\eta ,i}\times \omega _{\eta ,i+1},}\\ K_{ii}(z)&= {} \left[ \begin{array}{cccccccc} 0 &{} 0 &{} \cdots &{} 0 &{} 0 \\ z^{N_{\eta }}d_{\eta }^*(z) &{} 0 &{} \cdots &{} 0 &{} 0 \\ 0 &{} z^{N_{\eta }}d_{\eta }^*(z) &{} \cdots &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} z^{N_{\eta }}d_{\eta }(z) &{} 0 \end{array} \right] _{\omega _{\eta ,i}\times \omega _{\eta ,i}} \end{aligned}$$
and
$$\begin{aligned} K_{m_{\eta }m_{\eta }}(z)=\left[ \begin{array}{cccccccc} \frac{g_{\eta }}{\omega _{\eta ,m_{\eta }}}z^{N_{\eta }}d_{\eta }^{co}(z) &{} \frac{g_{\eta }}{\omega _{\eta ,m_{\eta }}}z^{N_{\eta }}d_{\eta }^{co}(z) &{} \cdots &{} \frac{g_{\eta }}{\omega _{\eta ,m_{\eta }}}z^{N_{\eta }}d_{\eta }^{co}(z) &{} \frac{g_{\eta }}{\omega _{\eta ,m_{\eta }}}z^{N_{\eta }}d_{\eta }^{co}(z)\\ z^{N_{\eta }}d_{\eta }^*(z) &{} 0 &{} \cdots &{} 0 &{} 0 \\ 0 &{} z^{N_{\eta }}d_{\eta }^*(z) &{} \cdots &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} z^{N_{\eta }}d_{\eta }(z) &{} 0 \end{array} \right] _{{\omega _{\eta ,m_{\eta }}\times \omega _{\eta ,m_{\eta }}.}} \end{aligned}$$
(2) The proof of Lemma 1. (i) \(K_{\eta }(z)\) can be derived directly from the definition of the matrixes \(K_k^{\eta }\). (ii) Substituting \(z=1\) into \(K_{\eta }(z)\) yields the matrix \(K_{\eta }(1)\). \(K_{\eta }(1)=\)
Since \(K_{\eta }(1)\) is irreducible stochastic, we can verify that \(z=1\) is a zero point of \(det\varDelta _{\eta }(z)\) by direct determinant calculation. (iii) From the equation \({\varvec{\xi }}_{\eta }={\varvec{\xi }}_{\eta }K_{\eta }(1)\), we have
$$\begin{aligned}&\left\{ \begin{array}{llll} \xi _{0k-1}^{\eta }=\xi _{0k}^{\eta }+\frac{g_{\eta }}{\omega _{\eta ,0}}(\xi _{00}^{\eta }+\xi _{10}^{\eta }+\cdots +\xi _{m_{\eta }0}^{\eta }) &{} \text{ for } k=1,\ldots ,\omega _{\eta ,0}-1, \\ \xi _{0\omega _{\eta ,0}-1}^{\eta }=\frac{g_{\eta }}{\omega _{\eta ,0}}(\xi _{00}^{\eta }+\xi _{10}^{\eta }+\cdots +\xi _{m_{\eta }0}^{\eta }), &{}\end{array} \right. \\&\left\{ \begin{array}{llll} \xi _{i+1k-1}^{\eta }=\xi _{i+1k}^{\eta }+\frac{1-g_{\eta }}{\omega _{\eta ,i+1}}\xi _{i0}^{\eta } &{} \text{ for } k=1,\ldots ,\omega _{\eta ,i+1}-1;\ i=0, \ldots m_{\eta }-2, \\ \xi _{i+1\omega _{\eta ,i+1}-1}^{\eta }=\frac{1-g_{\eta }}{\omega _{\eta ,i+1}}\xi _{i0}^{\eta }&{} \end{array} \right. \\&\left\{ \begin{array}{llll} \xi _{m_{\eta }k-1}^{\eta }=\xi _{m_{\eta }k}^{\eta }+\frac{1-g_{\eta }}{\omega _{\eta ,m_{\eta }}}(\xi _{m_{\eta }-10}^{\eta }+\xi _{m_{\eta }0}^{\eta }) &{} \text{ for } k=1,\ldots ,\omega _{\eta ,m_{\eta }}-1 \\ \xi _{m\omega _{m_{\eta }}-1}^{\eta }=\frac{1-g_{\eta }}{\omega _{\eta ,m_{\eta }}}(\xi _{m_{\eta }-10}^{\eta }+\xi _{m_{\eta }0}^{\eta }). &{} \end{array} \right. \end{aligned}$$
Using these relations, the formulae (4.8)–(4.10) can be obtained by induction. Finally, (4.11) can be established from the equation \({\varvec{\xi }}_{\eta }e_{\omega _{\eta }}=1\). This completes the proof.
(3) The proof of Lemma 2. The argument is similar to that given in [11] and [12]. We present it here for the paper self-contained. Denote the classical adjoin matrix of the kernel matrix \(\varDelta _{\eta }(z)\) by \(adj\varDelta _{\eta }(z)\), then \(\{adj \varDelta _{\eta }(z)\}\times \varDelta _{\eta }(z)=det\varDelta _{\eta }(z)I_{\omega _{\eta }}\). Differentiating the equation with respect to z, evaluating the result at \(z=1\) and multiplying on the right by \(e_{\omega _{\eta }}\) yield \(\{adj \varDelta _{\eta }(1)\}\varDelta _{\eta }'(1)e_{\omega _{\eta }}+\frac{d}{dz}\{adj\varDelta _{\eta }(z)\}|_{z=1}\varDelta _{\eta }(1)e_{\omega _{\eta }}=\gamma e_{\omega _{\eta }}.\) From \(\varDelta _{\eta }(1)e_{\omega _{\eta }}=0\), we obtain \(\{adj\varDelta _{\eta }(1)\}\)\([N_{\eta }I_{\omega _{\eta }}-K_{\eta }'(1)]e_{\omega _{\eta }}=\gamma _{\eta } e_{\omega _{\eta }}.\) Since \(\{adj\varDelta _{\eta }(1)\}\varDelta _{\eta }(1)=\mathbf{0}\), we further have \(adj\varDelta _{\eta }(1)=c[{\varvec{\xi }}_{\eta },\ldots ,{\varvec{\xi }}_{\eta }]^{\tau }\) where \(c>0\). Consequently, \(c[{\varvec{\xi }}_{\eta },\ldots ,{\varvec{\xi }}_{\eta }]^{\tau }[N_{\eta }I_{\omega _{\eta }}-K_{\eta }'(1)]e_{\omega _{\eta }}=\gamma _{\eta } e_{\omega _{\eta }}\), so that the claim (i) holds. To prove the claim (ii), note that under the assumption 1, \(\gamma _{\eta }>0\) holds. According to Theorem 6 in [12], the Markov chain is ergodic. The proof is completed.
Appendix 4
We prove Theorem 1 in two steps: (1) the calculation of the determinant \(det\varDelta _{\eta }(z)\) and (2) the discussion on the distribution situation of zeros of \(det\varDelta _{\eta }(z)\). Concretely, in the step 1, based on the structure characteristic of the matrix \(\varDelta _{\eta }(z)\), we calculate its determinant using the following formula of block matrix determinant iteratively
$$\begin{aligned} det \left( \begin{array}{ccc} A &{} B \\ C &{} D \end{array} \right) =det(A)det(D-CA^{-1}B) \end{aligned}$$
and in step 2, we discuss the zeros of \(det\varDelta _{\eta }(z)\) using Rouché’s theorem.
Step 1. The derivation of (i) of Theorem 1. Note that the first rows of all the sub-matrixes \(K_{i0}(z),\ i=1,\ldots ,m_{\eta }\) are same and non-zero. Then, for the determinant \(det\varDelta _{\eta }(z)\), multiplying the first row of \(-K_{m_{\eta }0}(z)\) by -1 and adding it to the first row of \(-K_{i0}(z),\ i=1,\ldots ,m_{\eta }-1\) yield
$$\begin{aligned}&det\varDelta _{\eta }(z)\\&\quad =\left| \begin{array}{ccccccccc} z^{N_{\eta }}I_{\omega _{\eta ,0}}-K_{00}(z) &{} -K_{01}(z) &{} \cdots &{} \mathbf{0} \\ \mathbf{0} &{} z^{N_{\eta }}I_{\omega _{\eta ,1}}-K_{11}(z) &{} \cdots &{} \mathbf{0} \\ \mathbf{0} &{} \mathbf{0} &{} \cdots &{} \mathbf{0} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \mathbf{0} &{} \mathbf{0} &{} \cdots &{} z^{N_{\eta }}I_{\omega _{\eta ,m_{\eta }-1}}-K_{m_{\eta }-1m_{\eta }-1}(z) \\ -K_{m_{\eta }0}(z) &{} \mathbf{0} &{} \cdots &{} \mathbf{0} \end{array} \right. \\&\quad \left. \begin{array}{ccccccccc} &{} \mathbf{0} \\ &{} -K^{(0)}_{1m_{\eta }}(z) \\ &{} -K^{(0)}_{2m_{\eta }}(z) \\ &{} \vdots \\ &{} -K^{(0)}_{m_{\eta }-1m_{\eta }}(z) \\ &{} z^{N_{\eta }}I_{\omega _{\eta ,m_{\eta }}}-K_{m_{\eta }m_{\eta }}(z) \end{array} \right| \end{aligned}$$
where
$$\begin{aligned}&-K^{(0)}_{im_{\eta }}(z)\\&=\left[ \begin{array}{cccccccc} z^{N_{\eta }}-\frac{g_{\eta }}{\omega _{\eta ,m}}z^{N_{\eta }}d_{\eta }^{co}(z) &{} -\frac{g_{\eta }}{\omega _{\eta ,m_{\eta }}}z^{N_{\eta }}d_{\eta }^{co}(z) &{} \cdots &{} -\frac{g_{\eta }}{\omega _{\eta ,m}}z^{N_{\eta }}d_{\eta }^{co}(z) \\ 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 \end{array} \right] _{\omega _{\eta ,i}\times \omega _{\eta ,m},} \end{aligned}$$
and
$$\begin{aligned} -K^{(0)}_{m_{\eta }-1m_{\eta }}(z)=\left[ \begin{array}{cccccccc} z^{N_{\eta }} &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 \end{array} \right] _{\omega _{\eta ,m_{\eta }-1}\times \omega _{\eta ,m_{\eta }}}. \end{aligned}$$
For \(i=1,\ldots ,m_{\eta }-2\), let
$$\begin{aligned} \varDelta _i^{\eta }(z)&= {} \left[ \begin{array}{ccccccccc} z^{N_{\eta }}I_{\omega _{\eta ,i}}-K_{ii}(z) &{} -K_{ii+1}(z) &{} \cdots &{} \mathbf{0} \\ \mathbf{0} &{} z^{N_{\eta }}I_{\omega _{\eta ,(i+1)}}-K_{i+1i+1}(z) &{} \cdots &{} \mathbf{0} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \mathbf{0} &{} \mathbf{0} &{} \cdots &{} z^{N_{\eta }}I_{\omega _{\eta ,m_{\eta }-1}}-K_{m_{\eta }-1m_{\eta }-1}(z) \\ \mathbf{0} &{} \mathbf{0} &{} \cdots &{} \mathbf{0} \end{array} \right. \\&\quad \left. \begin{array}{ccccccccc} &{} -K^{(0)}_{im_{\eta }}(z) \\ &{} -K^{(0)}_{i+1m_{\eta }}(z) \\ &{} \vdots \\ &{} -K^{(0)}_{m_{\eta }-1m_{\eta }}(z) \\ &{}z^{N_{\eta }}I_{\omega _{\eta ,m_{\eta }}}-K_{m_{\eta }m_{\eta }}(z)-\sum _{l=1}^{i}H_{m_{\eta }m_{\eta }}^{(l)}(z) \end{array} \right] . \end{aligned}$$
We have
$$\begin{aligned} det\varDelta _{\eta }(z)&= {} \left| \begin{array}{c|cccccccc} z^{N_{\eta }}I_{\omega _{\eta ,0}}-K_{00}(z) &{} -K_{01}(z) &{} \mathbf{0} &{} \cdots &{} &{} \mathbf{0} \\ \mathbf{0} &{} &{} &{} &{} \\ \vdots &{} &{} \varDelta _1^{\eta }(z) &{} &{} \\ \mathbf{0} &{} &{} &{} &{} \\ -K_{m_{\eta }0}(z) &{} &{} &{} &{} \end{array} \right| \\&=det(z^{N_{\eta }}I_{\omega _{\eta ,0}}-K_{00}(z))det (\varDelta _1^{\eta }(z)- \left[ \begin{array}{ccccccc} \mathbf{0} \\ \mathbf{0} \\ \vdots \\ \mathbf{0} \\ -K_{m_{\eta }0}(z) \end{array} \right] [z^{N_{\eta }}I_{\omega _{\eta ,0}}-K_{00}(z)]^{-1} \times \\&\quad {[}\ -K_{01}(z)\ \ \mathbf{0}\ \ \cdots \ \ \mathbf{0}\ \ \mathbf{0}\ ] ) \\&\quad \equiv det(z^{N_{\eta }}I_{\omega _{\eta ,0}}-K_{00}(z)) detD(z) \end{aligned}$$
We first calculate \(det(z^{N_{\eta }}I_{\omega _{\eta ,0}}-K_{00}(z))\) in the last equality by induction. Based on the structure of the matrix \(z^{N_{\eta }}I_{\omega _{\eta ,0}}-K_{00}(z)\), we have
$$\begin{aligned}&det(z^{N_{\eta }}I_{\omega _{\eta ,0}}-K_{00}(z))\\&\quad =\left| \begin{array}{cccccc} z^{N_{\eta }}-\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) &{} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) &{} \cdots &{} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) &{} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) \\ -z^{N_{\eta }}d^*_{\eta }(z) &{} z^{N_{\eta }} &{} \cdots &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} -z^{N_{\eta }}d_{\eta }(z) &{} z^{N_{\eta }} \end{array} \right| _{\omega _{\eta ,0}\times \omega _{\eta ,0}}\\&\quad =(z^{N_{\eta }}-\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z))\left| \begin{array}{ccccc} z^{N_{\eta }} &{} &{} &{} \\ -z^{N_{\eta }}d_{\eta }^*(z) &{} z^{N_{\eta }} &{} &{} \\ &{} \ddots &{} \ddots &{} \\ &{} &{} -z^{N_{\eta }}d_{\eta }(z) &{} z^{N_{\eta }} \end{array} \right| _{(\omega _{\eta ,0}-1)\times (\omega _{\eta ,0}-1)}\\&\quad -(-z^{N_{\eta }}d_{\eta }^*(z)) \left| \begin{array}{cccccc} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) &{} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) &{} \cdots &{} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) &{} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) \\ -z^{N_{\eta }}d_{\eta }^*(z) &{} z^{N_{\eta }} &{} \cdots &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{}\vdots \\ 0 &{} 0 &{} \cdots &{} -z^{N_{\eta }}d_{\eta }(z) &{} z^{N_{\eta }} \end{array} \right| \\&\quad =(z^{N_{\eta }}-\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z))z^{N_{\eta }(\omega _{\eta ,0}-1)}+\\&\quad z^{N_{\eta }}d_{\eta }^*(z)\left\{ -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z)\left| \begin{array}{ccccc} z^{N_{\eta }} &{} &{} &{} \\ -z^{N_{\eta }}d_{\eta }^*(z) &{} z^{N_{\eta }} &{} &{} \\ &{} \ddots &{} \ddots &{} \\ &{} &{} -z^{N_{\eta }}d_{\eta }(z) &{} z^{N_{\eta }} \end{array} \right| \right. \\&\quad \left. -(-z^{N_{\eta }}d_{\eta }^*(z)) \left| \begin{array}{cccccc} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) &{} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) &{} \cdots &{} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) &{} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) \\ -z^{N_{\eta }}d_{\eta }^*(z) &{} z^{N_{\eta }} &{} \cdots &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{}\vdots \\ 0 &{} 0 &{} \cdots &{} -z^{N_{\eta }}d_{\eta }(z) &{} z^{N_{\eta }} \end{array} \right| \right\} \\&\displaystyle =(z^{N_{\eta }}-\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z))z^{N_{\eta }(\omega _{\eta ,0}-1)}-\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z)z^{N_{\eta }(\omega _{\eta ,0}-1)}d_{\eta }^*(z)\\&\quad +(z^{N_{\eta }}d_{\eta }^*(z))^2\left| \begin{array}{cccccc} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) &{} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) &{} \cdots &{} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) &{} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) \\ -z^{N_{\eta }}d_{\eta }^*(z) &{} z^{N_{\eta }} &{} \cdots &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{}\vdots \\ 0 &{} 0 &{} \cdots &{} -z^{N_{\eta }}d_{\eta }(z) &{} z^{N_{\eta }} \end{array} \right| \\&\displaystyle =(z^{N_{\eta }}-\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z))z^{N_{\eta }(\omega _{\eta ,0}-1)}-\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z)z^{N_{\eta }(\omega _{\eta ,0}-1)}d_{\eta }^*(z)-\\&\quad \frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z)z^{N_{\eta }(\omega _{\eta ,0}-1)}d_{\eta }^{*2}(z)-\cdots \\&-\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z)z^{N_{\eta }(\omega _{\eta ,0}-1)}d_{\eta }^{*(\omega _{\eta ,0}-3)}(z)+\\&\quad (z^{N_{\eta }}d_{\eta }^*(z))^{\omega _{\eta ,0}-2} \left| \begin{array}{cccccc} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) &{} -\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z) \\ -z^{N_{\eta }}d_{\eta }(z) &{} z^{N_{\eta }} \end{array} \right| \\&\quad \displaystyle =z^{N_{\eta }(\omega _{\eta ,0}-1)}\left( z^{N_{\eta }}-\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z)\left( 1+d_{\eta }^*(z)+\cdots +d_{\eta }^{*(\omega _{\eta ,0}-2)}(z)d_{\eta }(z)\right) \right) . \end{aligned}$$
Next, we calculate the second term detD(z) by induction. Note that
$$\begin{aligned}&\left[ \begin{array}{ccccccc} \mathbf{0} \\ \mathbf{0} \\ \vdots \\ \mathbf{0} \\ -K_{m_{\eta }0}(z) \end{array} \right] [z^{N_{\eta }}I_{\omega _{\eta ,0}}-K_{00}(z)]^{-1}[\ -K_{01}(z)\ \ \mathbf{0}\ \ \cdots \ \ \mathbf{0}\ ]\\&\quad =\left[ \begin{array}{ccccccc} \mathbf{0} &{} \mathbf{0} &{} \cdots &{} \mathbf{0} &{} \mathbf{0}\\ \vdots &{} \vdots &{} \ddots &{} \vdots &{}\vdots \\ \mathbf{0} &{} \mathbf{0} &{} \cdots &{} \mathbf{0} &{} \mathbf{0}\\ H_{m_{\eta }1}^{(1)}(z) &{} \mathbf{0} &{} \cdots &{} \mathbf{0} &{} H_{m_{\eta }m_{\eta }}^{(1)}(z) \end{array} \right] \end{aligned}$$
with \(H_{m_{\eta }1}^{(1)}(z)=(-K_{m_{\eta }0}(z))[z^{N_{\eta }}I_{\omega _{\eta ,0}}-K_{00}(z)]^{-1}(-K_{01}(z))\) and \(H_{m_{\eta }m_{\eta }}^{(1)}(z)=\mathbf{0}\). This gives
$$\begin{aligned}&detD(z) = \left| \begin{array}{ccccccccc} z^{N_{\eta }}I_{\omega _{\eta ,1}}-K_{11}(z) &{} -K_{12}(z) &{} \cdots &{} \mathbf{0} \\ \mathbf{0} &{} z^{N_{\eta }}I_{\omega _{\eta ,2}}-K_{22}(z) &{} \cdots &{} \mathbf{0} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \mathbf{0} &{} \mathbf{0} &{} \cdots &{} z^{N_{\eta }}I_{\omega _{\eta ,m_{\eta }-1}}-K_{m_{\eta }-1m_{\eta }-1}(z)\\ H_{m_{\eta }1}^{(1)}(z) &{} \mathbf{0} &{} \cdots &{} \mathbf{0} \end{array} \right. \\&\quad \left. \begin{array}{cccc} -K^{(0)}_{1m_{\eta }}(z) \\ -K^{(0)}_{2m_{\eta }}(z) \\ \vdots \\ -K^{(0)}_{m_{\eta }-1m_{\eta }}(z) \\ z^{N_{\eta }}I_{\omega _{\eta ,m_{\eta }}}-K_{m_{\eta }m_{\eta }}(z) -H_{m_{\eta }m_{\eta }}^{(1)}(z) \end{array} \right| \\&\quad =det(z^{N_{\eta }}I_{\omega _{\eta ,1}}-K_{11}(z))det\\&\quad \left( \varDelta _2(z)-\left[ \begin{array}{ccccccc} \mathbf{0} \\ \mathbf{0} \\ \vdots \\ \mathbf{0} \\ -H_{m_{\eta }1}^{(1)}(z) \end{array} \right] [z^{N_{\eta }}I_{\omega _{\eta ,1}}-K_{11}(z)]^{-1} \times \right. \\&\left. [\ -K_{12}(z)\ \ \mathbf{0}\ \ \cdots \ \ \mathbf{0}\ \ -K_{1m_{\eta }}^{(0)}\ ] \right) \\&\quad =det(z^{N_{\eta }}I_{\omega _{\eta ,1}}-K_{11}(z))det\\&\quad \left( \varDelta _2(z)-\left[ \begin{array}{ccccccc} \mathbf{0} &{} \mathbf{0} &{} \cdots &{} \mathbf{0} &{} \mathbf{0}\\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ \mathbf{0} &{} \mathbf{0} &{} \cdots &{} \mathbf{0} &{} \mathbf{0} \\ H_{m_{\eta }1}^{(2)}(z) &{} \mathbf{0} &{} \cdots &{} \mathbf{0} &{} H_{m_{\eta }m_{\eta }}^{(2)}(z) \end{array} \right] \right) \\&\quad =det(z^{N_{\eta }}I_{\omega _{\eta ,1}}-K_{11}(z))\times \\&\quad det\left[ \begin{array}{ccccccccc} z^{N_{\eta }}I_{\omega _{\eta ,2}}-K_{22}(z) &{} -K_{23}(z) &{} \cdots &{} \mathbf{0} \\ \mathbf{0} &{} z^{N_{\eta }}I_{\omega _{\eta ,3}}-K_{33}(z) &{} \cdots &{} \mathbf{0} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \mathbf{0} &{} \mathbf{0} &{} \cdots &{} z^{N_{\eta }}I_{\omega _{\eta ,m_{\eta }-1}}-K_{m_{\eta }-1m_{\eta }-1}(z) \\ -H_{m_{\eta }2}^{(2)}(z) &{} \mathbf{0} &{} \cdots &{} \mathbf{0} \end{array} \right. \\&\quad \left. \begin{array}{ccccc} -K^{(0)}_{2m_{\eta }}(z) \\ -K^{(0)}_{3m_{\eta }}(z) \\ \vdots \\ -K^{(0)}_{m_{\eta }-1m_{\eta }}(z) \\ z^{N_{\eta }}I_{\omega _{\eta ,m_{\eta }}}-K_{m_{\eta }m_{\eta }}(z)-\sum _{i=1}^2H_{m_{\eta }m_{\eta }}^{(i)}(z) \end{array} \right] \end{aligned}$$
with \(H_{m_{\eta }2}^{(2)}(z) =(-H_{m_{\eta }1}^{(1)}(z))[z^{N_{\eta }}I_{\omega _{\eta ,1}}-K_{11}(z)]^{-1}(-K_{12}(z))\) and \(H_{m_{\eta }m_{\eta }}^{(2)}(z) =(-H_{m_{\eta }1}^{(1)}(z))[z^{N_{\eta }}I_{\omega _{\eta ,1}}-K_{11}(z)]^{-1}\times (-K_{1m_{\eta }}^{(0)}(z))\). The structure of the matrix of the second term in the last equality is the same as that of D(z). Thus, repeating the above procedure yields
$$\begin{aligned}&detD(z) = det(z^{N_{\eta }}I_{\omega _{\eta ,1}}-K_{11}(z))\cdots \\&\quad det(z^{N_{\eta }}I_{\omega _{\eta ,m_{\eta }-2}}-K_{m_{\eta }-2m_{\eta }-2}(z))\\&\quad \times det\left[ \begin{array}{ccccc} z^{N_{\eta }}I_{\omega _{\eta ,m_{\eta }-1}}-K_{m_{\eta }-1m_{\eta }-1}(z) &{} -K_{m_{\eta }-1m_{\eta }}^{(0)} \\ -H_{m_{\eta }m_{\eta }-1}^{(m_{\eta }-1)} &{} z^{N_{\eta }}I_{\omega _{\eta ,m}}-K_{m_{\eta }m_{\eta }}(z)-\sum _{i=1}^{m_{\eta }-1}H_{m_{\eta }m_{\eta }}^{(i)}(z) \end{array} \right] \\&\quad =det(z^{N_{\eta }}I_{\omega _{\eta ,1}}-K_{11}(z))det(z^{N_{\eta }}I_{\omega _{\eta ,2}}-K_{22}(z))\cdots \cdots \\&\quad det(z^{N_{\eta }}I_{\omega _{\eta ,m_{\eta }-1}}-K_{m_{\eta }-1m_{\eta }-1}(z))\\&\quad \times det\left( (z^{N_{\eta }}I_{\omega _{\eta ,m_{\eta }-1}}-K_{m_{\eta }m_{\eta }}(z) -\sum _{l=1}^{m_{\eta }-1}H_{m_{\eta }m_{\eta }}^{(l)}(z))-(-H_{m_{\eta }m_{\eta }-1}^{(m_{\eta }-1)}(z)) \right. \\&\quad \times \left. [z^{N_{\eta }}I_{\omega _{\eta ,m_{\eta }-1}}-K_{m_{\eta }-1m_{\eta }1-}(z))]^{-1}(-K_{m_{\eta }-1m_{\eta }}^{(0)}(z))\right) \\&\quad =\prod _{i=1}^{m_{\eta }-1}det(z^{N_{\eta }}I_{\omega _{\eta ,i}}-K_{ii}(z))\\&\quad \times det\left( z^{N_{\eta }}I_{\omega _{\eta ,m}}-K_{m_{\eta }m_{\eta }}(z)-\sum _{i=1}^{m_{\eta }}H_{m_{\eta }m_{\eta }}^{(i)}(z))\right) \end{aligned}$$
where \(H_{m_{\eta }m_{\eta }}^{(m_{\eta })}(z)=(-H_{m_{\eta }m_{\eta }-1}^{(m_{\eta }-1)}(z))[z^{N_{\eta }}I_{\omega _{\eta ,m_{\eta }-1}}-K_{m_{\eta }-1m_{\eta }1-}(z))]^{-1}(-K_{m_{\eta }-1m_{\eta }}^{(0)}(z))\). For \(i=1,\ldots ,m_{\eta }-1 \) we have
$$\begin{aligned} z^{N_{\eta }}I_{\omega _{\eta ,i}}-K_{ii}(z)=\left[ \begin{array}{cccccc} z^{N_{\eta }} &{} &{} &{} &{} \\ -z^{N_{\eta }}d^*_{\eta }(z) &{}z^{N_{\eta }} &{} &{} \\ &{} \ddots &{} \ddots &{} \\ &{} &{} -z^{N_{\eta }}d_{\eta }(z) &{} z^{N_{\eta }} \end{array} \right] _{\omega _i\times \omega _i,} \end{aligned}$$
so that \(det[z^{N_{\eta }}I_{\omega _{\eta ,i}}-K_{ii}(z)]=z^{N_{\eta }\omega _{\eta ,i}} \) and
$$\begin{aligned}&{[}z^{N_{\eta }}I_{\omega _{\eta ,i}}-K_{ii}(z)]^{-1}=z^{-N_{\eta }}\\&\quad \left[ \begin{array}{cccccc} 1 &{} &{} &{} &{} \\ d^*_{\eta }(z) &{} 1 &{} &{} \\ d^{*2}_{\eta }(z) &{} d^*_{\eta }(z) &{} \ddots &{} &{} \\ \vdots &{} \vdots &{} \ddots &{} \ddots \\ d^{*(\omega _{\eta ,i}-2)}(z)d_{\eta }(z) &{}d^{*(\omega _{\eta ,i}-3)}(z)d_{\eta }(z) &{}\cdots &{} d_{\eta }(z) &{} 1 \end{array} \right] \end{aligned}$$
This gives \(\prod _{i=1}^{m_{\eta }-1}det(z^{N_{\eta }}I_{\omega _{\eta ,i}}-K_{ii}(z))=z^{N_{\eta }(\omega _{\eta ,1}+\cdots +\omega _{\eta ,m_{\eta }-1})}\). The factor \(det(z^{N_{\eta }}I_{\omega _{\eta ,m}}-K_{m_{\eta }m_{\eta }}(z)-\sum _{i=1}^{m_{\eta }}H_{m_{\eta }m_{\eta }}^{(i)}(z))\) can be calculated as follows. Recall that all the elements of the matrixes \(K_{m_{\eta }0}(z), K_{ii+1}(z)\) and \(K_{im_{\eta }}^{(0)}(z)\) are zero except for the first rows. This property is also inherited by all the matrices \(H_{m_{\eta }m_{\eta }}^{(i)}(z),\ i=1,\ldots , m_{\eta }\). Thus, we can write \(H_{m_{\eta }m_{\eta }}(z)\equiv \sum _{i=1}^{m_{\eta }}H_{m_{\eta }m_{\eta }}^{(i)}(z)\) as
$$\begin{aligned} H_{m_{\eta }m_{\eta }}(z)=\left[ \begin{array}{cccccc} H_{m_{\eta }m_{\eta }}(1,1)(z) &{} H_{m_{\eta }m_{\eta }}(1,2)(z) &{} \cdots &{} H_{m_{\eta }m_{\eta }}(1,\omega _{\eta ,m_{\eta }})(z) \\ 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 \end{array} \right] . \end{aligned}$$
We have
$$\begin{aligned}&z^{N_{\eta }}I_{\omega _{\eta ,m_{\eta }}}-K_{m_{\eta }m_{\eta }}(z)-H_{m_{\eta }m_{\eta }}(z)\\&\quad =\left[ \begin{array}{ccccccc} z^{N_{\eta }}-\frac{z^{N_{\eta }}(1-g_{\eta })d_{\eta }^{co}}{\omega _{\eta ,m_{\eta }}}-H_{m_{\eta }m_{\eta }}(1,1) &{} -\frac{z^{N_{\eta }}(1-g_{\eta })d_{\eta }^{co}}{\omega _{\eta ,m_{\eta }}}-H_{m_{\eta }m_{\eta }}(1,2) &{} \cdots \\ -z^{N_{\eta }}d_{\eta }^*(z) &{} z^{N_{\eta }} &{} \cdots \\ &{} \ddots &{} \ddots \\ &{} &{} \end{array} \right. \\&\quad \left. \begin{array}{ccccc} -\frac{z^{N_{\eta }}(1-g_{\eta })d_{\eta }^{co}}{\omega _{\eta ,m_{\eta }}}-H_{m_{\eta }m_{\eta }}(1,\omega _{\eta ,m_{\eta }}-1)&{} -\frac{z^{N_{\eta }}(1-g_{\eta })d_{\eta }^{co}}{\omega _{\eta ,m_{\eta }}}-H_{m_{\eta }m_{\eta }}(1,\omega _{\eta ,m_{\eta }}) \\ 0 &{} 0 \\ \vdots &{} \vdots \\ -z^{N_{\eta }}d_{\eta }(z) &{} z^{N_{\eta }} \end{array} \right] \end{aligned}$$
As seen, this matrix has the same structure as the matrix \(z^{N_{\eta }}I_{\omega _{\eta ,0}}-K_{00}(z)\). By similar argument we obtain the determinant
$$\begin{aligned}&det(z^{N_{\eta }}I_{\omega _{\eta ,m_{\eta }}}-K_{m_{\eta }m_{\eta }}(z)-\sum _{i=1}^{m_{\eta }}H_{m_{\eta }m_{\eta }}^{(i)}(z))\\&\quad =z^{(\omega _{\eta ,m_{\eta }}-1)N_{\eta }}\left( z^{N_{\eta }}-\frac{1-g_{\eta }}{\omega _{\eta ,m_{\eta }}}z^{N_{\eta }}d_{\eta }^{co}(z)\right. \\&\quad (\sum _{l=1}^{\omega _{\eta ,m_{\eta }}-1}d_{\eta }^{*(l-1)}(z)+d_{\eta }^{*(\omega _{\eta ,m_{\eta }}-1)}(z)d_{\eta }(z))- \\&\quad (\sum _{l=1}^{\omega _{\eta ,m_{\eta }}-1}H_{m_{\eta }m_{\eta }}(1,l)(z)d_{\eta }^{*(l-1)}(z)\\&\quad \left. + H_{m_{\eta }m_{\eta }}(1,\omega _{\eta ,m_{\eta }})(z)d_{\eta }^{*(\omega _{\eta ,m_{\eta }}-1)}d_{\eta }(z))\right) . \end{aligned}$$
Substituting the above results into detD(z), we finally get the expression
$$\begin{aligned} detD(z)&= {} z^{N_{\eta }(\omega _{\eta }-\omega _{\eta ,0}-1)}\\&\quad \left( z^{N_{\eta }}-\frac{1-g_{\eta }}{\omega _{\eta ,m_{\eta }}}z^{N_{\eta }}d_{\eta }^{co}(z)(\sum _{l=1}^{\omega _{\eta ,m_{\eta }}-1}d_{\eta }^{*(l-1)}(z)+d_{\eta }^{*(\omega _{\eta ,m_{\eta }}-1)}(z)d_{\eta }(z))\right. \\&\quad -\left( \sum _{l=1}^{\omega _{\eta ,m_{\eta }}-1}H_{m_{\eta }m_{\eta }}(1,l)(z)d_{\eta }^{*(l-1)}(z)\right. \\&\quad \left. \left. + H_{m_{\eta }m_{\eta }}(1,\omega _{\eta ,m_{\eta }})(z)d_{\eta }^{*(\omega _{\eta ,m_{\eta }}-1)}d_{\eta }(z)\right) \right) . \end{aligned}$$
where \(\omega _{\eta }=\omega _{\eta ,0}+\omega _{\eta ,1}+\cdots +\omega _{\eta ,m_{\eta }}\). This completed the proof of (i).
Step 2. The proof of (ii) and (iii) of Theorem 1. For the claim (ii), define \(f_1(z)=z^{N_{\eta }}\) and \(f_2(z)=-\frac{g_{\eta }}{\omega _{\eta ,0}}d_{\eta }^{(N_{\eta })}(z)(1+d_{\eta }^*(z)+d_{\eta }^{*2}(z)+\cdots +d_{\eta }^{*(\omega _{\eta ,0}-2)}(z)d_{\eta }(z))\), then both \(f_1(z)\) and \(f_2(z)\) are analytic in \(|z|<1\) and continuous on \(|z|\le 1\). For z with \(|z|=1\),
$$\begin{aligned} |f_2(z)|&= {} \frac{g_{\eta }}{\omega _{\eta ,0}}|d_{\eta }^{(N_{\eta })}(z)||1+d_{\eta }^*(z)+d_{\eta }^{*2}(z)+\cdots \\&\quad +d_{\eta }^{*(\omega _{\eta ,0}-2)}(z)d_{\eta }(z)| \\&\le \frac{g_{\eta }}{\omega _{\eta ,0}}|d_{\eta }^{(N_{\eta })}(z)|(1+|d_{\eta }^*(z)|+|d_{\eta }^{*}(z)|^2+\cdots \\&\quad +|d_{\eta }^{*}(z)|^{\omega _{\eta ,0}-2}|d_{\eta }(z)|)\\&\le g_{\eta }<1=|z|^{N_{\eta }}=|f_1(z)|. \end{aligned}$$
An application of Rouché’s theorem shows that \(a_{\eta }(z)=f_1(z)+f_2(z)\) and \(f_1(z)\) have the same number of zeroes in the open unit disk. Next, we prove the claim (iii). Let
$$\begin{aligned} f_1(z)&= {} z^{N_{\eta }}\left( 1-\frac{1-g_{\eta }}{\omega _{\eta ,m_{\eta }}}d_{\eta }^{co}(z)\right. \\&\quad \left. \left( \sum _{l=1}^{\omega _{\eta ,m_{\eta }}-1}d_{\eta }^{*(l-1)}(z)+d_{\eta }^{*(\omega _{\eta ,m_{\eta }}-1)}(z)d_{\eta }(z)\right) \right) , \\ f_2(z)&= {} \sum _{l=1}^{\omega _{\eta ,m_{\eta }}-1}H_{m_{\eta }m_{\eta }}(1,l)(z)d_{\eta }^{*(l-1)}(z)\\&\quad + H_{m_{\eta }m_{\eta }}(1,\omega _{\eta ,m_{\eta }})(z)d_{\eta }^{*(\omega _{\eta ,m_{\eta }}-1)}(z)d_{\eta }(z), \end{aligned}$$
and let \(b_t(z)=f_1(z)+tf_2(z)\) for \(0\le t <1\). Then \(f_1(z)\) and \(f_2(z)\), thus \(b_t(z)\) are analytic in \(|z|<1\) and continuous on \(|z|\le 1\). Furthermore, \(b_t(z)\) is also a continuous function of t. First we determine the number and location of zeroes of \(b_t(z)\) for \(0\le t<1\). Since for any z in \(|z|\le 1\),
$$\begin{aligned}&\left| 1-\frac{1-g_{\eta }}{\omega _{\eta ,m_{\eta }}}d_{\eta }^{co}(z)(\sum _{l=1}^{\omega _{\eta ,m_{\eta }}-1}d_{\eta }^{*(l-1)}(z)+d_{\eta }^{*(\omega _{\eta ,m_{\eta }}-1)}(z)d_{\eta }(z))\right| \\&\ge 1-\left| \frac{1-g_{\eta }}{\omega _{\eta ,m_{\eta }}}d_{\eta }^{co}(z)(\sum _{l=1}^{\omega _{\eta ,m_{\eta }}-1}d_{\eta }^{*(l-1)}(z)+d_{\eta }^{*(\omega _{\eta ,m_{\eta }}-1)}(z)d_{\eta }(z))\right| \\&\ge 1-\frac{1-g_{\eta }}{\omega _{\eta ,m_{\eta }}}d_{\eta }^{co}(1)(\sum _{l=1}^{\omega _{\eta ,m_{\eta }}-1}d_{\eta }^{*(l-1)}(1)+d_{\eta }^{*(\omega _{\eta ,m_{\eta }}-1)}(1)d_{\eta }(1))=g_{\eta }>0, \end{aligned}$$
the factor \(1-\frac{1-g_{\eta }}{\omega _{\eta ,m_{\eta }}}d_{\eta }^{co}(z)(\sum _{l=1}^{\omega _{\eta ,m_{\eta }}-1}d_{\eta }^{*(l-1)}(z)+d_{\eta }^{*(\omega _{\eta ,m_{\eta }}-1)}(z)d_{\eta }(z))\) has no zeroes in \(|z|\le 1\). Hence \(f_1(z)\) has exactly \(N_{\eta }\) zeroes in the open unit disk and \(|f_1(z)|\ge g_{\eta }\) for z with \(|z|=1\). According to Lemma 1, \(z_{2N_{\eta }}=1\) is a zero point of \(det\varDelta _{\eta }(z)\), we have \(det\varDelta _{\eta }(1)=a_{\eta }(1)b_{\eta }(1)=0\). Due to \(a_{\eta }(1)=g_{\eta }\ne 0\), it must hold that \(b_{\eta }(1)=0\), which gives \(f_2(1)=-f_1(1)=-g_{\eta }\). Thus for \(|z|=1\) and \(0\le t <1\), we have \(|tf_2(z)|\le t(\sum _{l=1}^{\omega _{\eta ,m_{\eta }}-1}|H_{m_{\eta }m_{\eta }}(1,l)(z)||d_{\eta }^{*}(z)|^{l-1}+ |H_{m_{\eta }m_{\eta }}(1,\omega _{\eta ,m_{\eta }})(z)||d_{\eta }^{*}(z)|^{\omega _{\eta ,m_{\eta }}-1}|d_{\eta }(z)|\ ) \le tf_2(1)=tg_{\eta }<|f_1(z)|.\) Again according to Rouché’s theorem, we can see that \(b_t(z)=f_1(z)+tf_2(z)\) has exactly \(N_{\eta }\) zeroes (counting multiplicities) in the open unit disk and no zeroes on \(|z|=1\). Next, we consider \(b_{\eta }(z)\) and prove that under Assumption 1, there exists a unique zero \(z_{2N_{\eta }}(t)\) of \(b_t(z)\) such that \(\lim _{t\rightarrow 1}z_{2N_{\eta }}(t)=1=z_{2N_{\eta }}\). From \(\gamma _{\eta }=\frac{d}{dz}det\varDelta _{\eta }(z)|_{z=1}=a_{\eta }(1)b_{\eta }'(1)\), we have \(b_{\eta }'(1)=\gamma _{\eta }/g_{\eta }>0\). For z near 1, the first Taylor series expansion of \(b_t(z)\) gives
$$\begin{aligned} b_t(z)=b_t(1)+\gamma _t(z-1)+\epsilon _t(z)=g\gamma _t(1-t)+\gamma _t(z-1)+\epsilon _t(z) \end{aligned}$$
where \(\gamma _t=b'_t(1) \rightarrow b'(1)\) as \(t\rightarrow 1\), and \(\epsilon _t(z)/(z-1)\rightarrow 0\) as \(z\rightarrow 1\) uniformly in \(0\le t<1\). Define \(l_t(z)=g_{\eta }(1-t)+\gamma _t(z-1)\), and note that \(l_t(z)\) has one zero \(z_t=1-\frac{g_{\eta }(1-t)}{\gamma _t}\) which is near but less than 1 when t is near 1. Let \(\varOmega (\delta )=\{z:\ |z|<1, |z-1|<\delta \}\). We claim that \(|\epsilon _t(z)|=|b_t(z)-l_t(z)|<|l_t(z)|\) when \(z\in \partial \varOmega (\delta )\) for \(\delta \) sufficiently small and t near 1. Choose \(\delta \le \frac{1}{2}\) and \(t_{\delta }<1\) such that, for \(t_{\delta }<t<1\), \(\epsilon _t(z)/(z-1)|<\gamma /8g_{\eta }\) for \(|z-1|\le \delta ,\ \gamma _t>\gamma /2g_{\eta }\) and \(b_t(1)=g_{\eta }(1-t)< \delta /4g_{\eta }\). On the part of the boundary of \(\varOmega (\delta )\) where \(|z-1|=\delta \), it holds that for \(t_{\delta }<t<1\),
$$\begin{aligned} |l_t(z)|\ge \gamma _t|z-1|-g_{\eta }(1-t)>\frac{\gamma _{\eta }}{4g_{\eta }}|1-z|>|\epsilon _t(z)|. \end{aligned}$$
On the part of \(\partial \varOmega (\delta )\) where \(|z|=1\), since \(|z-1|^2\le 2|y|^2\) we have \(|l_t(z)|=\)
$$\begin{aligned}&|g_{\eta }(1-t)+\gamma _t(z-1)|=|g_{\eta }(1-t)+i\gamma _ty+\gamma _t(x-1)|\ge |g_{\eta }(1-t)+i\gamma _ty|-\gamma _t(1-x)\\&\ge \frac{g_{\eta }(1-t)}{\sqrt{2}}+\frac{\gamma _t|y|}{\sqrt{2}}-\frac{\gamma _t|z-1|^2}{2}\ge \gamma _t\frac{|z-1|}{2}(1-\delta )\ge \frac{\gamma }{4g_{\eta }}|z-1|>|\epsilon _t(z)|. \end{aligned}$$
Again from Rouché’s theorem, we get that \(b_t(z)\) and \(l_t(z)\) have the same number of zeroes in \(\varOmega (\delta )\), that is, just one zero (denoted by \(z_{2N_{\eta }}(t)\)). Obviously, \(\lim _{t\rightarrow 1}z_{2N_{\eta }}(t)=1\). The proof is completed.
Appendix 5
The derivation of Theorem 2. The idea is that firstly, obtain the power series expansion of \({\varvec{\pi }}_{\eta }(z)e_{\omega _{\eta }}\) at \(z=1\) by using the eigenvalues and eigenvectors of \(K_{\eta }(z)\), and then derive the Eq. (4.14) by substituting \(z=1\) into this power series expansion. Denote the eigenvalues of \(K=K_{\eta }(1)\) by \(x_n\), \(n=1,\ldots ,\omega _{\eta }\), especially, \(x_1=1\). Let \(x_n(z)\) be the analytic eigenvalue function of K(z), and \(\mathbf{u}_n(z)\) and \(\mathbf{v}_n(z)\) the corresponding analytic left and right eigenvectors in a neighborhood of \(z=1\), respectively. The power series expansion of those functions and vectors at \(z=1\) are given as follows.
$$\begin{aligned} x_n(z)&= {} x_n+(z-1)x_n^{(1)}+\frac{1}{2}(z-1)^2x_n^{(2)}+o((z-1)^2), \\ \mathbf{u}_n(z)&= {} \mathbf{u}_n+(z-1)\mathbf{u}_n^{(1)}+\frac{1}{2}(z-1)^2\mathbf{u}_n^{(2)}+o((z-1)^2), \\ \mathbf{v}_n(z)&= {} \mathbf{v}_n+(z-1)\mathbf{v}_n^{(1)}+\frac{1}{2}(z-1)^2\mathbf{v}_n^{(2)}+o((z-1)^2). \end{aligned}$$
From (4.7) we have
$$\begin{aligned}&{\varvec{\pi }}_{\eta }(z)e_{\omega _{\eta }}\nonumber \\&\quad =\left( {\varvec{\pi }}_0^{\eta }(A_{\eta }(z)-I_{\omega _{\eta }})+\sum _{l=1}^{N_{\eta }-1}{\varvec{\pi }}_l^{\eta }(B_{\eta }(z)-zI_{\omega _{\eta }})z^l\right) (z^{N_{\eta }}I_{\omega _{\eta }}-K_{\eta }(z))^{-1}e_{\omega _{\eta }}. \end{aligned}$$
(A.1)
Directly extending the first term of the right hand at \(z=1\) yields
$$\begin{aligned}&{\varvec{\pi }}_0^{\eta }(A_{\eta }(z)-I_{\omega _{\eta }})+\sum _{l=1}^{N_{\eta }-1}{\varvec{\pi }}_l^{\eta }(B_{\eta }(z)-zI_{\omega _{\eta }})z^l\nonumber \\&\quad ={\varvec{\pi }}_0^{\eta }(A_{\eta }-I_{\omega _{\eta }})+\sum _{l=1}^{N_{\eta }-1}{\varvec{\pi }}_l^{\eta }(B_{\eta }-I_{\omega _{\eta }})+(z-1)\nonumber \\&\quad \left[ {\varvec{\pi }}_0^{\eta }A_{\eta }'+\sum _{l=1}^{N_{\eta }-1}{\varvec{\pi }}_l^{\eta }(B_{\eta }'-I_{\omega _{\eta }})+\right. \nonumber \\&\quad \left. \sum _{l=1}^{N_{\eta }-1}l{\varvec{\pi }}_l^{\eta }(B_{\eta }-I_{\omega _{\eta }})\right] \nonumber \\&\quad +\frac{1}{2}(z-1)^2\left[ {\varvec{\pi }}_0^{\eta }A_{\eta }''+\sum _{l=1}^{N_{\eta }-1}{\varvec{\pi }}_l^{\eta }B_{\eta }''+2\sum _{l=1}^{N_{\eta }-1}l{\varvec{\pi }}_l^{\eta }(B_{\eta }'-I_{\omega _{\eta }})+\right. \nonumber \\&\quad \left. \sum _{l=2}^{N_{\eta }-1}l(l-1){\varvec{\pi }}_l^{\eta }(B_{\eta }-I_{\omega _{\eta }})\right] +o((z-1)^2). \end{aligned}$$
(A.2)
Next we consider the expansion of \((z^{N_{\eta }}I_{\omega _{\eta }}-K_{\eta }(z))^{-1}\). From the definitions of \(\mathbf{u}_n\) and \(\mathbf{v}_n\) we see that
$$\begin{aligned} (z^{N_{\eta }}I_{\omega _{\eta }}-K_{\eta }(z))^{-1} =\mathbf{V}(z)(z^{N_{\eta }}I_{\omega _{\eta }}-\varTheta (\mathbf{x}(z)))^{-1}\mathbf{U}(z) \end{aligned}$$
(A.3)
with \(\mathbf{U}(z)=(\mathbf{u}_1(z),\ldots ,\mathbf{u}_{\omega _{\eta }}(z))\), \(\mathbf{V}(z)=(\mathbf{v}_1^{\tau }(z),\ldots ,\mathbf{v}^{\tau }_{\omega _{\eta }}(z))^{\tau }\), \(\mathbf{x}(z)=(x_1(z), \ldots ,x_{\omega _{\eta }}(z))\) and \(\varTheta (\mathbf{x}(z))\) denotes the diagonal matrix whose the nth diagonal element is \(x_n(z)\). Let \(\mathbf{y}(z)=(\frac{1}{z^{N_{\eta }}-x_1(z)}, \ldots , \frac{1}{z^{N_{\eta }}-x_{\omega _{\eta }}(z)})\equiv (y_1(z),\ldots , y_{\omega _{\eta }}(z))\). Note that \(z=1\) is the pole of \(y_1(z)\), so the expansion of \(y_n(z)\) is given by
$$\begin{aligned} y_n(z)=\left\{ \begin{array}{lllll} \frac{1}{(z-1)(N_{\eta }-x_1^{(1)})}-\frac{N_{\eta }(N_{\eta }-1)-x_1^{(2)}}{2(N-x_1^{(1)})^2}+o(1), n=1,\\ \frac{1}{1-x_n}+o(1), n\ge 2. \end{array} \right. \end{aligned}$$
(A.4)
Moreover,
$$\begin{aligned} \begin{array}{lll} \mathbf{V}(z)=V+(z-1)V^{(1)}+\frac{1}{2}(z-1)^2V^{(2)}+o((z-1)^2), \\ \mathbf{U}(z)e_{\omega _{\eta }}=\mathbf{1}_1+(z-1){\varvec{\beta }}+o(z-1) \end{array} \end{aligned}$$
(A.5)
where \(\mathbf{1}_1=(1,0,\ldots ,0)^{\tau }\) and \({\varvec{\beta }}=(\beta _1,\ldots ,\beta _{\omega _{\eta }})\) with \(\beta _1=0\) and \(\beta _n=\mathbf{u}_n^{(1)}e_{\omega _{\eta }}=\mathbf{u}_nK_{\eta }'e_{\omega _{\eta }}/(1-x_n)\) for \(n\ge 2\). Substituting the expressions (A.2)-(A.5) into (A.1) and using the fact that
$$\begin{aligned} \begin{array}{llll} \mathbf{V}(z)\mathbf{1}_1=\mathbf{1}_1+(z-1)\mathbf{v}_1^{(1)}+\frac{1}{2}(z-1)^2\mathbf{v}_1^{(2)}+o((z-1)^2), \\ \mathbf{V}(z){\varvec{\beta }}=\sum _{n=2}^{\omega }v_n\beta _n+o(1), \end{array} \end{aligned}$$
we obtain the power series expansion of \({\varvec{\pi }}_{\eta }(z)e_{\omega _{\eta }}\) at \(z=1\).
$$\begin{aligned}&{\varvec{\pi }}_{\eta }(z)e_{\omega _{\eta }}\\&\quad =\frac{1}{N_{\eta }-x_1^{(1)}}\left\{ {\varvec{\pi }}_0^{\eta }\left[ (A_{\eta }-I_{\omega _{\eta }})\mathbf{v}_1^{(1)}+A_{\eta }'e_{\omega _{\eta }}\right] +\sum _{l=1}^{N_{\eta }-1}{\varvec{\pi }}_l^{\eta }\right. \\&\quad \left. \left[ (B_{\eta }-I_{\omega _{\eta }})\mathbf{v}_1^{(1)}+(B_{\eta }'-I_{\omega _{\eta }})e_{\omega _{\eta }}\right] \right\} \\&\quad +\frac{z-1}{N_{\eta }-x_1^{(1)}}\left\{ \left[ {\varvec{\pi }}_0^{\eta }(A_{\eta }-I_{\omega _{\eta }})+\sum _{l=1}^{N_{\eta }-1}{\varvec{\pi }}_l^{\eta }(B_{\eta }-I)\right] \left[ \mathbf{v}_1^{(2)}-\frac{1}{2}\frac{N_{\eta }(N_{\eta }-1)-x_1^{(2)}}{N_{\eta }-x_1^{(1)}}\mathbf{v}_1^{(1)}- \right. \right. \\&\quad \left. \left. \sum _{n=2}^{\omega _{\eta }}\frac{(N_{\eta }-x_1^{(1)})\mathbf{u}_nK_{\eta }'e_{\omega _{\eta }}}{(1-x_n)^2}{} \mathbf{v}_n\right] \right. -\frac{N_{\eta }(N_{\eta }-1)-x_1^{(2)}}{2(N_{\eta }-x_1^{(1)})}\left[ {\varvec{\pi }}_0^{\eta }A_{\eta }'+\sum _{l=1}^{N_{\eta }-1}{\varvec{\pi }}_l^{\eta }(B_{\eta }'-I_{\omega _{\eta }})\right] e_{\omega _{\eta }}\\&\quad +\left[ {\varvec{\pi }}_0^{\eta }A_{\eta }'+\sum _{l=1}^{N_{\eta }-1}{\varvec{\pi }}_l^{\eta }(B_{\eta }'-I_{\omega _{\eta }})+\sum _{l=1}^{N_{\eta }-1}l{\varvec{\pi }}_l^{\eta }(B_{\eta }-I_{\omega _{\eta }})\right] \mathbf{v}_1^{(1)}\\&\quad \left. +\frac{1}{2}\left[ {\varvec{\pi }}_0^{\eta }A_{\eta }''+\sum _{l=1}^{N_{\eta }-1}{\varvec{\pi }}_lB_{\eta }''+2\sum _{l=1}^{N_{\eta }-1}l{\varvec{\pi }}_l^{\eta }(B_{\eta }'-I_{\omega _{\eta }})+\sum _{l=2}^{N_{\eta }-1}l(l-1){\varvec{\pi }}_l^{\eta }(B_{\eta }-I_{\omega _{\eta }})\right] \mathbf{v}_1\right\} +o((z-1)) \end{aligned}$$
where \(x_1=1,\ x_1^{(1)}={\varvec{\xi }}K_{\eta }'e_{\omega _{\eta }}\), \(\mathbf{v}_1=e_{\omega _{\eta }}\) and \(\mathbf{u}_1={\varvec{\xi }}\). Finally, substituting \(z=1\) into the above expansion and noting that \({\varvec{\pi }}_{\eta }(1)e_{\omega _{\eta }}=1\), we establish the Eq. (4.16). This completes the proof.
