An expository note on Prohorov metric and Prohorov Theorem

TL;DR

This paper provides an exposition of weak convergence, Prohorov theorem, and Prohorov spaces, exploring their relationships and conditions for convergence of probability measures in various metric space settings.

Contribution

It offers a detailed explanation of Prohorov metric, theorem, and spaces, including new insights into their interrelations and convergence criteria in different metric spaces.

Findings

Relationship between Levy and Prohorov distances analyzed

Conditions for weak convergence and tightness established

Prohorov theorem proved in various metric space contexts

Abstract

The main aim of this article is to give an exposition of weak convergence, Prohorov theorem and Prohorov spaces. In this context we study the relationship between Levy distance $\ell(F, G)$ between two distribution functions $F$ and $G$ and the Prohorov distance $\pi( \mu, \nu)$ between the probability measures $\mu$ and $\nu$ determined by $F$ and $G$ respectively. We study the relationship among the weak convergence of probability measures ($\mu_n$) determined by distribution functions ($F_n$) to the probability measure $\mu$ determined by a distribution function $G$, the convergence of $\ell(F_n, G)$ and $\pi( \mu_n, \nu)$ to zero under suitable assumptions on the metric space on which these measures are defined. Tightness of probability measures and relative sequential compactness are studied and Prohorov theorem is proved in different settings. Prohorov spaces and non-Prohorov spaces are discussed.