4.1 Definition and properties
Definition 4.1.1. A discrete process \(\{Z_t\}\) is purely random if it consists of a sequence of mutually independent random variables, all with the same distribution, with mean \(\mu \) and variance \(\sigma _Z^2\).
Proposition 4.1.2. A purely random process is strictly stationary, with \[ \gamma (k) = \begin {cases} \sigma _Z^2 , & k = 0 , \\ 0 , & k = \pm 1, \pm 2, \dots , \end {cases} \qquad \rho (k) = \begin {cases} 1 , & k = 0 , \\ 0 , & k = \pm 1, \pm 2, \dots \end {cases} \]
Proof. For any times \(t_1, \dots , t_n\), the joint distribution of \(Z_{t_1}, \dots , Z_{t_n}\) is the product of \(n\) copies of the common distribution, so it is unchanged by a shift. Independent variables are uncorrelated, so \(\gamma (k) = 0\) for \(k \ne 0\), and \(\gamma (0) = \sigma _Z^2\). □
A purely random process is often called white noise; strictly, “white noise” only requires the variables to be uncorrelated with constant mean and variance, which is enough for most of the theory. If the \(Z_t\) are also normal, independence and uncorrelatedness coincide, and the process is Gaussian white noise. White noise has no memory at all: knowing past values does not help to predict future ones. It is important not for itself but as the building block from which the moving average, autoregressive and ARMA processes of the following sections are constructed, usually with \(\mu = 0\). Its correlogram is the benchmark for a series with no structure: the sample autocorrelations are approximately \(N(0, 1/n)\), as noted in Section 2.5.
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