Advanced Probability


results in ergodic theory as presented in Chapter 2 can be found in Breiman (1968) and Loève (1977b). A major application of ergodic theory is to the theory of the class of stochastic processes known as Markov processes. In Markov process theory, stationary processes are frequently encountered, and thus the theory presents an opportunity for utilizing the ergodic theorem for stationary processes. Basic introductions to Markov processes in both discrete and continuous time can be found in Norris (1999) and Brémaud (1999). In Meyn & Tweedie (2009), a more general theory is presented, which includes a series of results on ergodic Markov processes. In Chapter 3, we dealt with weak convergence. In its most general form, weak convergence of probability measures can be cast in the context of probability measures on complete, separable metric spaces, where the metric space considered is endowed with the Borel-σalgebra generated by the open sets. A classical exposition of this theory is found in Billingsley (1999), with Parthasarathy (1967) also being a useful resource.


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