Convergence Vague (IA) - Suites de Vecteurs Al\'eatoires

TL;DR

This monograph provides a comprehensive overview of weak convergence theory for random vectors, emphasizing applications in asymptotic statistics and including tools like the empirical process and specific convergence techniques.

Contribution

It offers a detailed, self-contained presentation of weak convergence in $ ^k$, including specialized tools and applications in asymptotic statistical analysis.

Findings

Detailed weak convergence criteria for random vectors

Application of empirical processes in asymptotic problems

Tools for handling convergence with i.i.d. sequences

Abstract

This monograph aims at presenting the core weak convergence theory for sequences of random vectors with values in $\mathbb{R}^k$. In some places, a more general formulation in metric spaces is provided. It lays out the necessary foundation that paves the way to applications in particular subfields of the theory. In particular, the needs of Asymptotic Statistics are addressed. A whole chapter is devoted to weak convergence in $\mathbb{R}$ where specific tools, for example for handling weak convergence of sequences using independent and indentically distributed random variables such that the Renyi's representations by means of standard uniform or exponential random variables, are stated. The function empirical process is presented as a powerful tool for solving a considerable number of asymptotic problems in Statistics. The text is written in a self-contained approach whith the proofs of all used results at the exception of the general Skorohod-Wichura Theorem