As the title reads, why is the convariance function for a stationary gaussian process not 0?
Isn't every realization X(t) independent of the other realizations? If not, how come?
Thanks.
2026-03-26 03:00:47.1774494047
Why is the convariance function for a stationary gaussian process not 0?
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No, it's not independent. In fact, think of two points very close in time, for example $X(t+h)$ and $X(t)$ for $h$ small. You can see that for a continuous Gaussian process the value of $X(t+h)$ will be very correlated to the one of $X(t)$. The covariance function indicates "how strong" this correlation is. For a zero-mean Gaussian process $X(t)$, the covariance $$ C(t, s) = E(X(t)X(s)), $$ is therefore a measure of how strong this correlation is. This is independent of the notion of stationarity.