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?

**Answer**

Let Xt be a zero-mean covariance-stationary time series such that

Xt=φ1Xt−1+φ2Xt−2+εt

where εt is white noise.

Using L to mean the lag (backshift) operator, the above can be expressed as

(1−φ1L−φ2L2)Xt=εt.

Since Xt is a covariance-stationary AR(2) process, the roots of its characteristic polynomial (1−φ1z−φ2z2)=0 must lie outside the unit circle. Thus, Equation (1) can be written as

(1−λ1L)(1−λ2L)Xt=εt

where |λ1|<1 and |λ2|<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/λ1 and z∗2=1/λ2, will lie outside the unit circle. Therefore,

Xt=1(1−λ1L)1(1−λ2L)εt.

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).

**Attribution***Source : Link , Question Author : Monolite , Answer Author : JoOkuma*