Let

describe a stationary AR(p) process, where is a white noise process with zero mean and variance , and is a sequence of AR coefficients. Given , , ..., which are taken to be uncorrelated Gaussian deviates with zero mean and unit variance (obtained on a computer from a Gaussian random number generator), we desire to generate a realization of , , ..., . To do so, we carry out the following steps.

1. We first calculate the p-1 sequences , , ..., and by computing the following for k = p, p-1, ..., 2:

   

2. Second, we calculate

   

3. Third, we generate , , ..., via

   

4. Finally, the remaining N-p values are generated using

   

Let us consider two concrete examples, namely, the AR(2) and AR(4) processes given by Equations (45) and (46a). The AR(2) process has coefficients and and has . Application of step 1 yields

while step 2 yields

We thus would generate the AR(2) process using

For the AR(4) process, we have , , and , with . Application of step 1 yields

while step 2 yields