convergence in distribution ConvergenceInDistribution 2002-12-10 07:50:49 2005-02-11 12:36:21 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A sequence of distribution functions $F_1,F_2,\dots$ converges \emph{weakly} to a distribution function $F$ if $F_n(t)\rightarrow F(t)$ for each point $t$ at which $F$ is continuous. If the random variables $X,X_1,X_2,\dots$ have associated distribution functions $F,F_1,F_2,\dots$, respectively, then we say that $X_n$ converges \emph{in distribution} to $X$, and denote this by $X_n\xrightarrow[]{D} X$. This definition holds for joint distribution functions and random vectors as well. This is probably the weakest \PMlinkescapetext{type} of convergence of random variables. Some results involving this \PMlinkescapetext{type} of convergence are the central limit theorems, Helly-Bray theorem, Paul L\'evy continuity theorem, Cram\'er-Wold theorem and Scheff\'e's theorem.