A Study of Weak Convergence of Probability Measures with an Overview of Empirical Processes

Dey, Anirban (2021) A Study of Weak Convergence of Probability Measures with an Overview of Empirical Processes. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS Dissertation of Anirban Dey (16MS151)) 16MS151_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (754kB)

Abstract

Weak convergence is the notion of convergence in distribution generalised to arbitrary metric spaces. Important theorems describing methods to establish weak convergence and implications of such convergence are reviewed in the first chapter. Thereafter, the theory is developed in the special case of C[0; 1] with topics including Brownian motion, Donsker's theorem, arc since laws etc. The Skorohod topology for D[0; 1] is most convenient to study convergence of cadlag processes and derive results similar to C[0; 1]. Notable among the cases where the weak convergence of D comes into play signifi- cantly is studying the asymptotic properties of empirical distributions and empirical processes. The classical empirical process is viewed as [0; 1] or R- indexed stochastic process, but the indexing set can be extended to a class of functions F through a simple logical argument. Such F-indexed empirical processes find extensive use in modern statistical applications such as Mestimation, Z-estimation etc. and thus, form an active area of research; the second half of the report is dedicated entirely to their study. To address measurability issues, a more generalised desnition of weak convergence becomes necessary. After discussing various modes of convergence for non-measurable maps, we proceed to study empirical process theory in greater detail. Various GLivenko-Cantelli and Donsker results are mentioned, Vapnik-Cervonenkis classes are described with entropy bounds. The final section summarises a few results for statistical applications such as M-estimation and goodness-of- fit testing.

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