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2: The Estimation of Mean and Covariances

In this brief second chapter, some results concerning asymptotic properties of the sample mean and the sample ACVF are collected. Throughout, \((X_t\colon t\in\mathbb{Z})\) denotes a weakly stationary stochastic process with mean \(\mu\) and ACVF \(\gamma\). In Section 1.2 it was shown that such a process is completely characterized by these two quantities. The mean \(\mu\) was estimated by the sample mean \(\bar{x}\), and the ACVF \(\gamma\) by the sample ACVF \(\hat{\gamma}\) defined in (1.2.1). In the following, some properties of these estimators are discussed in more detail.