1. Elements of Stochastic Processes

A great deal of data in economics, finance, engineering, and the natural sciences occur in the form of time series where observations are dependent and where the nature of this dependence is of interest. A model which describes the probability structure of time series observations \(X_t\), \(t=1, 2,\ldots , n\), is called a stochastic process. An \(X_t\) might be the value of a stock return at time t, the water level in a river at time t, and so on. In this chapter, we will introduce a variety of stochastic processes, e.g., stationary processes, linear processes, heavy-tailed processes, etc., and explain some of their elements, e.g., stationarity, spectral analysis, prediction, causality, etc. The primary purpose of this book is to provide statistical inference, Whittle estimation, model diagnostics, portmanteau tests, nonstandard testing problems for the boundary parameter, and causality tests for heavy-tailed stochastic processes. This relies on the asymptotic theory for higher order asymptotics, Bartlett adjustment, empirical likelihood method, asymptotic efficiency, robustness, etc. For these problems, some useful fundamentals and tools will be given. Throughout this book, we denote by \(\mathbb {N}\), \(\mathbb {Z}\), and \({\mathbb {R}}^m\), the set of all positive integers, the set of all integers, and the m-dimensional Euclidean space, respectively. Stochastic processes were born as a mathematical model describing random quantities which vary together with time. For each time \(t\in \mathbb {Z}\), there exists a random variable \(X_t\) defined on a probability space \((\Omega , \mathscr {A}, P)\), then the family of random variables \(\{X_t: t\in \mathbb {Z}\}\) is called a stochastic process. From the definition, \(\{X_t\}\) may be a family of any random variables. But, if we want to do mathematical or statistical analysis, a sort of regularity or invariance for \(\{X_t\}\) is needed. The most fundamental one is stationarity.