Kolmogorov's strong law of large numbersKolmogorovsStrongLawOfLargeNumbers2002-12-08 05:39:292006-09-28 02:51:10Theorem% 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 hereLet $X_1,X_2,\dots$ be a sequence of independent random variables, with finite expectations. The strong law of large numbers holds if one of the following conditions is satisfied:
\begin{enumerate}
\item The random variables are identically distributed;
\item For each $n$, the variance of $X_n$ is finite, and
\[\sum_{n=1}^\infty \frac{\operatorname{Var}[X_n]}{n^2} <\infty.\]
\end{enumerate}