3.1 Random Process

A random process is a series of random variables indexed by time or space, \(\{Y_t\}_{t\in T}\), defined over a common probability space. We can organize a finite set of these variables in a random vector,

\[\mathbf{Y} = \left(\begin{array}{c} Y_{1}\\ Y_{2}\\ \vdots\\ Y_{m}\\ \end{array}\right)\]

• For time series data, the set of indices \(T=\{0,1,2,...,n\}\) indicates the time ordering of the random variables.

• For longitudinal data, we index our data for one unit/subject with a random process where \(T = \{1,2,...,m\}\) indicates the ordering of the repeated random variables over time.

• For spatial data, \(T\) refers to a set of points in space measured by (longitude, latitude).

Key Point: A random process can capture how a characteristic evolves over time (or space or both), and \(Y\) represents the random value of that characteristic.