设多项式\(A(z)\)的互异根是\(z_1,z_2,..,z_k\). 取\(1< \rho < \min\{|z_j|\}\). \(A^{-1}(z)=1/A(z)\) 是\(\{z: |z|\leq \rho \}\)内的解析函数. 从而有Taylor级数 \[\begin{align} A^{-1}(z) = \sum_{j=0}^\infty \psi_j z^j, \ |z| \leq \rho. \tag{8.7} \end{align}\] 由级数(8.7)在\(z=\rho\) 的收敛性得到 \(|\psi_j \rho^j|\to 0\), 当\(j\to \infty\). 于是由 \[\begin{align} \psi_j= o(\rho^{-j}), \ \hbox{ 当} \ j\to \infty \tag{8.8} \end{align}\] 知道\(\{\psi_j\}\)绝对可和. 而且, \(\min\{|z_j|\}\)越大, \(\psi_j\)趋于零越快.

令 \[\begin{aligned} A^{-1}(\mathscr B)=\sum_{j=0}^\infty \psi_j \mathscr B^j. \end{aligned}\] 如果\(\{X_t\}\)是(8.6)的平稳解， 则由定理7.2知 \[\begin{align} X_t = A^{-1}(\mathscr B) A(\mathscr B) X_t = A^{-1}(\mathscr B)\varepsilon_t, \ \ t\in \mathbb Z. \tag{8.9} \end{align}\] 可见平稳解如果存在必然为 \[\begin{align} X_t=A^{-1}(\mathscr B)\varepsilon_t =\sum_{j=0}^\infty \psi_j \varepsilon_{t-j}, \quad t\in \mathbb Z, \tag{8.10} \end{align}\] \(\{\psi_j\}\)称为平稳序列\(\{X_t\}\)的Wold系数。 显然(8.10)是线性平稳列， 将其代入模型(8.6)可以验证其满足模型， 所以\(X_t = A^{-1}(\mathscr B) \varepsilon_t\)是模型(8.6)的唯一的平稳解 （唯一是a.s.意义下）。

定理8.1 (1) 由(8.10)定义的时间序列\(\{X_t\}\) 是AR(\(p\))模型(8.5)的唯一(a.s.意义)平稳解；

(2) AR(\(p\))的模型的通解有如下形式 \[\begin{align} Y_t= \sum_{j=0}^\infty \psi_j\varepsilon_{t-j} + \sum_{j=1}^k \sum_{l=0}^{r(j)-1} U_{l,j} t^l z_j^{-t}, \quad t\in \mathbb Z. \tag{8.11} \end{align}\]

记\(a_0=-1\)则\(A(z)=-\sum_{j=0}^p a_j z^j\)， \[\begin{aligned} 1 = A(z) A^{-1}(z) = -\sum_{m=0}^\infty \left(\sum_{j=0}^p a_j \psi_{m-j}\right) z^m \end{aligned}\] 故\(\psi_0=1\), \(\sum_{j=0}^p a_j \psi_{m-j} = 0\), \(m>0\)。 于是 \[\begin{aligned} \begin{cases} \psi_0 = 1, \\ \psi_m = \sum_{j=1}^p a_j \psi_{m-j}, & m=1, 2, \dots \\ \psi_m =0, & m