The AR(3) process is stationary if the roots of its characteristic equation lie inside the unit circle¹². This is equivalent to the following conditions being met³:

1. $$\phi_1 + \phi_2 + \phi_3 < 1$$
2. $$\phi_2 - \phi_1 - \phi_3 < 1$$
3. $$\phi_3 > -1$$

Here, $\phi_1$, $\phi_2$, and $\phi_3$ are the coefficients of the AR(3) process³. These conditions form a region in the parameter space of $(\phi_1, \phi_2, \phi_3)$, and as long as the parameters lie within this region, the AR(3) process is stationary¹².

Source: Conversation with Bing, 10/7/2023
(1) time series - Stationarity of AR(p) process - Cross Validated. https://stats.stackexchange.com/questions/402696/stationarity-of-arp-process.
(2) Lecture 13 Time Series: Stationarity, AR(p) & MA(q) - Bauer College of .... https://www.bauer.uh.edu/rsusmel/phd/ec2-3.pdf.
(3) What are the stationarity conditions for an AR(4) process?. https://bing.com/search?q=stationarity+condition+for+AR%283%29+process.
(4) What are the stationarity conditions for an AR(4) process?. https://stats.stackexchange.com/questions/518136/what-are-the-stationarity-conditions-for-an-ar4-process.
(5) Whether a AR (P) process is stationary or not? - Cross Validated. https://stats.stackexchange.com/questions/19788/whether-a-arp-process-is-stationary-or-not.