Discussion
Let \(\{X_{t}\}\) be a stationary time series with autocovariance function \(r(h)\). Then, which of the following statements is not true about \(r(h)\) ?
Answer: C
The statement that is not true is \(|\gamma(k)| \ge \gamma(0)\).
By the Cauchy-Schwarz inequality, \(|\gamma(k)| = |\mathrm{Cov}(X_{t}, X_{t+k})|\), which is at most \(\sqrt{\mathrm{Var}X_{t}\,\mathrm{Var}X_{t+k}} = \gamma(0)\). So the true inequality is the reverse, \(|\gamma(k)| \le \gamma(0)\).
The others are standard properties of an autocovariance function: \(\gamma(0)\) is the variance (A), \(\gamma\) is even (B), and \(\gamma\) is non-negative definite (D), because \(\sum\sum \alpha_{i}\alpha_{j}\gamma(|t_{i} - t_{j}|)\) equals \(\mathrm{Var}(\sum \alpha_{i}X_{t_{i}}) \ge 0\).
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