CramerWoldTheorem 2002-12-10 08:49:40 2002-12-10 08:49:40 Theorem % 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 Let \[\overline{X}_n = (X_{n1},\dots,X_{nk}) \;\mbox{and} \; \overline{X} = (X_1,\dots,X_k)\] be \PMlinkescapetext{k-dimensional} random vectors. Then $\overline{X}_n$ converges to $\overline{X}$ \PMlinkname{in distribution}{ConvergenceInDistribution} if and only if \[\sum_{i=1}^k t_iX_{ni} \xrightarrow[n\rightarrow\infty]{D} \sum_{i=1}^k t_iX_i.\] for each $(t_1,\dots,t_k)\in \mathbb{R}^k$. That is, if every \PMlinkescapetext{fixed} linear combination of the coordinates of $\overline{X}_n$ converges in distribution to the correspondent linear combination of coordinates of $\overline{X}$.