AutocovarianceFunction

The autocovariance function of a stochastic process $X_t$ is a function that gives the covariance of the process with itself at pairs of time points. For a given process $X_t$, it's defined for all pairs of times $(s, t)$ as:

$$R(s, t) = E[(X_s - E[X_s])(X_t - E[X_t])]$$

In words, $R(s, t)$ is the expected product of the deviations of $X_s$ and $X_t$ from their expected values. This function measures how much $X_s$ and $X_t$ tend to vary together.

Here, $E[X]$ denotes the expected value of $X$. The process is assumed to have a well-defined mean and covariance function.

In the case of a zero-mean process, the autocovariance simplifies to:

$$R(s, t) = E[X_s X_t]$$

Because the expected value of a zero-mean process is 0, so $E[X_s] = E[X_t] = 0$.