Elements of Stochastic Processes

Abstract

Much of statistical analysis is concerned with models in which the observations are assumed to vary independently. However, a great deal of data in economics, 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 in itself. A model which describes the probability structure of a series of observations X t , t = 1,…, n, is called a stochastic process. An Xt might be the value of a stock price at time point t, the water level in a lake at time point t, and so on. The primary purpose of this book is to provide statistical inference for stochastic processes, which is based on the probability theory for them. In this chapter some of the elements of stochastic processes will be reviewed. Because the statistical analysis for stochastic processes largely relies on the asymptotic theory, we also explain some useful limit theorems and central limit theorems. We have placed some fundamental results of mathematics, probability, and statistics in the Appendix.