3 Stochastic processes

Time series are modelled as random processes. A stochastic process is a family of random variables \(X(\omega , t)\) indexed by time \(t\), where \(\omega \) belongs to a sample space and \(t\) to an index set (here the integers). For a fixed \(t\), \(X(\omega , t)\) is a random variable. For a fixed \(\omega \), \(X(\omega , t)\) as a function of \(t\) is called a sample function or a realisation. The population of all possible realisations is called the ensemble. An observed time series is thus one realisation of a stochastic process, and the aim of the analysis is to learn about the process from that single realisation. We write \(X_t\) for \(X(\omega , t)\).

A process is described completely by the joint distributions of every finite set of its variables: for any times \(t_1, \dots , t_n\), the joint distribution function \(F(x_{t_1}, \dots , x_{t_n})\) of \(X_{t_1}, \dots , X_{t_n}\). This is usually far more than can be known, and a simpler description uses the first and second moments:

• the mean function \(\mu (t) = E(X_t)\);
• the variance function \(\sigma ^2(t) = \Var (X_t) = E[(X_t - \mu (t))^2]\);
• the autocovariance function \(\gamma (t_1, t_2) = E[(X_{t_1} - \mu (t_1))(X_{t_2} - \mu (t_2))] = \Cov (X_{t_1}, X_{t_2})\).

The variance function is the special case \(\sigma ^2(t) = \gamma (t, t)\).

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