RandomWalk 2005-01-31 20:14:39 2006-10-22 14:52:22 Definition simple random walk symmetric simple random walk % 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,amscd} \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 \textbf{Definition}. Let $(\Omega,\mathcal{F},\mathbf{P})$ be a probability space and $\lbrace X_i \rbrace$ a discrete-time stochastic process defined on $(\Omega,\mathcal{F},\mathbf{P})$, such that the $X_i$ are iid real-valued random variables, and $i\in\mathbb{N}$, the set of natural numbers. The \emph{random walk} defined on $X_i$ is the sequence of partial sums, or partial series $$S_n\colon=\sum_{i=1}^{n}X_i.$$ If $X_i\in\lbrace -1,1 \rbrace$, then the random walk defined on $X_i$ is called a \emph{simple random walk}. A \emph{symmetric simple random walk} is a simple random walk such that $\mathbf{P}(X_i=1)=1/2$. The above defines random walks in one-dimension. One can easily generalize to define higher dimensional random walks, by requiring the $X_i$ to be vector-valued (in $\mathbb{R}^n$), instead of $\mathbb{R}$. \textbf{Remarks}. \begin{enumerate} \item Intuitively, a random walk can be viewed as movement in space where the length and the direction of each step are random. \item It can be shown that, the limiting case of a random walk is a Brownian motion (with some conditions imposed on the $X_i$ so as to satisfy part of the defining conditions of a Brownian motion). By limiting case we mean, loosely speaking, that the lengths of the steps are very small, approaching 0, while the total lengths of the walk remains a constant (so that the number of steps is very large, approaching $\infty$). \item If the random variables $X_i$ defining the random walk $w_i$ are integrable with zero mean $\operatorname{E}[X_i]=0$, $S_i$ is a martingale. \end{enumerate}