An AR(3) process is a stationary process if and only if the roots of the characteristic polynomial are outside the unit circle. The characteristic polynomial of an AR(3) process is given by:

```
1 - phi_1 * z - phi_2 * z^2 - phi_3 * z^3 = 0
```

where phi_1, phi_2, and phi_3 are the autoregressive parameters.

To check if the roots of the characteristic polynomial are outside the unit circle, we can use the following steps:

1. Calculate the roots of the characteristic polynomial using a numerical root-finding method.
2. Check if the absolute value of all the roots is greater than 1. If it is, then the process is stationary. Otherwise, the process is non-stationary.

For example, the following Python code checks if an AR(3) process is stationary:

```python
import numpy as np

def is_stationary(phi):
  """
  Checks if an AR(3) process is stationary.

  Args:
    phi: A list of three floats representing the AR coefficients.

  Returns:
    True if the process is stationary, False otherwise.
  """

  # Calculate the roots of the characteristic polynomial.
  roots = np.roots([1] + phi)

  # Check if the absolute value of all the roots is greater than 1.
  return np.all(np.abs(roots) > 1)

# Example usage:

phi = [0.5, 0.25, 0.125]
print(is_stationary(phi)) # True
```

In this example, the AR(3) process is stationary because the absolute value of all the roots of the characteristic polynomial is greater than 1.

It is important to note that the stationarity conditions for an AR(3) process are the same as the stationarity conditions for any AR(p) process, where p is the order of the AR process.

----------------

Yes, the stationarity condition for an AR(3) process can be expressed as a set of inequalities on the parameters of the process.

To do this, we can use the Routh-Hurwitz criterion. The Routh-Hurwitz criterion is a necessary and sufficient condition for the roots of a polynomial to be all in the left half-plane.

For an AR(3) process, the Routh-Hurwitz criterion is given by the following inequalities:

```
1 - phi_1 > 0
1 + phi_1 > 0
1 + phi_2 - phi_1^2 > 0
1 - phi_3 > 0
|1 + phi_1 - phi_2| > |phi_3|
```

where phi_1, phi_2, and phi_3 are the autoregressive parameters.

If all of these inequalities are satisfied, then the AR(3) process is stationary. Otherwise, the process is non-stationary.

For example, the following Python code checks if an AR(3) process is stationary using the Routh-Hurwitz criterion:

```python
import numpy as np

def is_stationary_using_routh_hurwitz(phi):
  """
  Checks if an AR(3) process is stationary using the Routh-Hurwitz criterion.

  Args:
    phi: A list of three floats representing the AR coefficients.

  Returns:
    True if the process is stationary, False otherwise.
  """

  # Check if all of the Routh-Hurwitz inequalities are satisfied.
  return (1 - phi_1 > 0 and
          1 + phi_1 > 0 and
          1 + phi_2 - phi_1^2 > 0 and
          1 - phi_3 > 0 and
          np.abs(1 + phi_1 - phi_2) > np.abs(phi_3))

# Example usage:

phi = [0.5, 0.25, 0.125]
print(is_stationary_using_routh_hurwitz(phi)) # True
```

In this example, the AR(3) process is stationary because all of the Routh-Hurwitz inequalities are satisfied.

I hope this helps!