The Probabilistic Structure of Time Series
Tucker McElroy, Dimitris N. Politis · 2019
This chapter discusses time series more formally, in the language of probability theory. It focuses on the mathematical foundations for random vectors and stochastic processes with the goal of introducing the important concept of stationarity. Typical time series datasets are an observed sample, say X1,X2,…,Xn, i.e., they constitute a finite stretch of a realization of the time series of interest. Hence, it is useful to recall certain facts about random vectors. A useful way of assessing and measuring properties of random variables and random vectors is to consider their expectation (also called the mean) – Appendix A for background on the expectation operator E. Higher order moments are typically needed to characterize the properties of non-Gaussian processes. The first and second moment structure have a special form when the time series is strictly stationary.