# How to write an AR(2) stationary process in the Wold representation

I managed to write an AR(1) process in the Wold representation with help from the geometric series.

I am having trouble with a stationary AR(2). How could I do?

Let $$XtX_t$$ be a zero-mean covariance-stationary time series such that
$$Xt=φ1Xt−1+φ2Xt−2+εtX_t = \varphi_1 X_{t-1} + \varphi_2 X_{t-2} + \varepsilon_t$$
where $$εt\varepsilon_t$$ is white noise.

Using $$LL$$ to mean the lag (backshift) operator, the above can be expressed as
$$(1−φ1L−φ2L2)Xt=εt.(1-\varphi_1L - \varphi_2L^2)X_t=\varepsilon_t . \tag{1}$$

Since $$XtX_t$$ is a covariance-stationary AR(2) process, the roots of its characteristic polynomial $$(1−φ1z−φ2z2)=0(1-\varphi_1 z - \varphi_2 z^2) = 0$$ must lie outside the unit circle. Thus, Equation (1) can be written as
$$(1−λ1L)(1−λ2L)Xt=εt(1-\lambda_1 L)(1-\lambda_2 L)X_t=\varepsilon_t$$
where $$|λ1|<1\vert \lambda_1 \rvert<1$$ and $$|λ2|<1\vert \lambda_2 \rvert<1$$. The last two inequalities are true of a covariance-stationary AR(2) process since then the roots of the characteristic polynomial, $$z∗1=1/λ1z^*_1=1/ \lambda_1$$ and $$z∗2=1/λ2z^*_2=1/ \lambda_2$$, will lie outside the unit circle. Therefore,
$$Xt=1(1−λ1L)1(1−λ2L)εt.X_t= \frac{1}{(1-\lambda_1 L)} \frac{1}{(1- \lambda_2 L)} \varepsilon_t .$$

Expand the two fractions on the right-hand side in the equation above using the geometric series. and you'll have the Wold decomposition of an AR(2).