Stationary Process

Recall the definition of Weak Stationary Process. If \(X_t\) satisfies \(\mu_X(t)\equiv C\), and \(r_X(t,t+\tau)=r_X(\tau)\), then we call \(X_t\) weak stationary process, or briefly stationary process.

Properties of self-relevant coefficient

Properties of self-relevant coefficient

(i) \(r_X(0)=\sigma^2\), and \(|r_X(\tau)|\leq r_X(0)\).

(ii) symmetry. \(r_X(-\tau)=\overline{r_X(\tau)}\).

(iii) \(r_X(\tau)\) is not negative definitive, i.e. \(\forall n\), \(a_1,\cdots,a_n\) and \(t_1,\cdots,t_n\),

\[ \sum_{k=1}^n \sum_{m=1}^n r_X(t_k-t_m) a_k\overline{a_m}\geq 0. \]

Here we dig into its mean value of time.