# Existence of weakly stationary process for given mean and covariance

I know that if a process is weakly stationary, the mean of the process will be time-independent and the covariance will be a function of the time difference.

My question: if I have a time-independent mean and a covariance, which is a function of the time difference, can I always construct a weakly stationary process (with those mean and covariance function)?

Kindly cite appropriate material, if possible. Thanks!

## 1 Answer

So long as you have a mean $$\mu$$ and an auto-covariance function $$\gamma$$, then for any $$n \in \mathbb{N}$$ you can define the observable vector $$\mathbf{X} = (X_1,...,X_n)$$ by:

$$\mathbf{X} \sim \text{N} ( \mu \mathbf{1}, \mathbf{\Sigma} ) \quad \quad \quad \mathbf{\Sigma} \equiv \begin{bmatrix} \gamma(0) & \gamma(1) & \cdots & \gamma(n-1) \\ \gamma(1) & \gamma(0) & \cdots & \gamma(n-2) \\ \vdots & \vdots & \ddots & \vdots \\ \gamma(n-1) & \gamma(n-2) & \cdots & \gamma(0) \\ \end{bmatrix}.$$

This can be extended to arbitrary $$n$$ so it is sufficient to form the stationary process $$\{ X_t | t \in \mathbb{Z} \}$$ according to the designated mean and auto-covariance function.