Abstract
In this chapter, we present some basic results from the theory of stochastic processes and investigate the properties of some standard continuous-time stochastic processes. In Sect. 1.1, we give the definition of a stochastic process. In Sect. 1.2, we present some properties of stationary stochastic processes. In Sect. 1.3, we introduce Brownian motion and study some of its properties. Various examples of stochastic processes in continuous time are presented in Sect. 1.4. The Karhunen–Loeve expansion, one of the most useful tools for representing stochastic processes and random fields, is presented in Sect. 1.5. Further discussion and bibliographical comments are presented in Sect. 1.6. Section 1.7 contains exercises.
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