# Autocorrelation function $\rho(s)$ of AR(p), when s goes infinity

Let $$\{X_t\}_{t\in\mathbb{Z}}$$ is the stacionary autoregressive process of degree p (AR(p)), and autocorrelation function of AR(p) is $$\rho(s)=\phi_1\rho(s-1)+\phi_2\rho(s-2)+\dots+\phi_p\rho(s-p), \text{ for s=1,2,\dots, p}.$$ I should show that when $$s\rightarrow \infty$$ then $$\rho(s)\rightarrow0$$. I am trying to solve difference equation and then $$s\rightarrow \infty$$. But I have no idea how to make it in general, because I think that when $$s$$ goes to the infinity ($$s\rightarrow \infty$$), it means also that p goes to the infinity ($$p\rightarrow \infty$$). Any help will be appreciated, thank you very much.

This can be seen by solving for $$\{\rho(s)\}_{s \geq 0}$$ directly.

The system of linear homogeneous difference equation (of degree $$p$$) is $$\underbrace{ \rho(s)- \phi_1\rho(s-1)+\phi_2\rho(s-2)+\dots+\phi_p\rho(s-p) }_{ \Phi(L)\rho(s)} = 0, \;\; s \geq {0}.$$ with initial conditions $$\rho(0), \cdots \rho(p-1)$$ given by the Yule-Walker equations

$$\begin{bmatrix} 1 &\rho(1) & \rho(2) &\cdots& \rho(p-1) \\ \rho(1) & 1 & \rho(1) &\cdots & \rho(p-2) \\ \rho(2) & \rho(1) & 1 &\cdots & \rho(p-3) \\ \vdots & \vdots & \vdots &\ddots & \vdots \\ \rho(p-1) & \rho(p-2) &\rho(p-3)&\cdots & 1 \\ \end{bmatrix} \begin{bmatrix} \phi_1 \\ \phi_2 \\ \phi_3 \\ \vdots \\ \phi_p \\ \end{bmatrix} = \begin{bmatrix} \rho(1) \\ \rho(2) \\ \rho(3) \\ \vdots \\ \rho(p) \\ \end{bmatrix}.$$

The general solution is a linear combination of terms corresponding to the AR polynomial $$\Phi(z)$$.

For example, if $$\Phi$$ have $$p$$ distinct real roots $$r_1, \cdots, r_{p}$$, the general solution is $$\rho(s) = c_1 r_1^{-s} + \cdots c_{p} r_{p}^{-s}.$$ The coefficients $$c_1, \cdots, c_{p}$$ are given by the initial conditions. By assumption (covariance stationarity), $$|r_1|, \cdots, |r_{p+1}| > 1$$, $$\lim_{s \rightarrow \infty} r_l^{-s} = 0, \; l = 1, \cdots, p.$$ This implies $$\lim_{s \rightarrow \infty} \rho(s) = 0.$$

The case of repeated or complex roots is similar. Same result holds.

The key condition is that all roots of $$\Phi$$ lies outside the unit circle. If there is a root on or inside the unit circle, e.g. if $$r =1$$ is a root, then the term $$1^{-s}$$ in $$\rho(s)$$ clearly does not approach $$0$$.