2.1 Second Order Stationary

A process is second order stationary or (weakly) if its mean is constant and its auto co variance function depends only on its lag i.e \(E(X_t)= \mu \) and \(Cov(x_t,X_{t+k})= \varphi (k)\)

if \(\tau = 0\) it implies that the mean and variance is constant.

Note. The variance and the mean must be finite.
The auto correlation coefficients are important for describing time series.
Suppose \(X_t\) has mean \(\mu \) and variance \(\sigma ^2\) acv.f \(\varphi (k)\), and ac.f \(\rho (\tau )\).

Then \begin {align*} \rho (\tau )&= \frac {\varphi (\tau )}{\varphi (\tau )}\\ &= \frac {\varphi (\tau )}{\sigma ^2} \end {align*}

Note. \(\rho (0)= 1\), also the ac.f is an even function \(\rho (\tau )= \rho (-\tau )\).
\(\implies (X_t,X_{t+\tau })= (X_t,X_{t-\tau })\)

Note.

1.: \begin {align*} \varphi (\tau )&= \rho (\tau )\,\cdot \,\sigma ^2\\ &= Cov(X_t,X_{t+\tau })\\ &= Cov(X_{t-\tau },X_t)\\ &= V(-\tau )\\ \end {align*}
2.: \(|\rho (\tau )|<1\)
3.: The ac.f lacks uniqueness. Several processes can have the same auto co variance.

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