In this chapter, we study concepts and theory useful in understanding the limiting behavior of stochastic processes. We begin with a general discussion of stochastic processes in metric spaces. The focus of this discussion is on measurable stochastic processes since most limits of empirical processes in statistical applications are measurable. We next discuss weak convergence both in general and in the specific case of bounded stochastic processes. One of the interesting aspects of the approach we take to weak convergence is that the processes studied need not be measurable except in the limit. This is useful in applications since many empirical processes in statistics are not measurable with respect to the uniform metric. The final section of this chapter considers other modes of convergence, such as in probability and outer almost surely, and their relationships to weak convergence.
KeywordsWeak Convergence Outer Probability Vector Lattice Sample Path Empirical Process
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