random walk
Definition. Let $(\mathrm{\Omega},\mathcal{F},\mathbf{P})$ be a probability space^{} and $\{{X}_{i}\}$ a discretetime stochastic process defined on $(\mathrm{\Omega},\mathcal{F},\mathbf{P})$, such that the ${X}_{i}$ are iid realvalued random variables^{}, and $i\in \mathbb{N}$, the set of natural numbers. The random walk^{} defined on ${X}_{i}$ is the sequence of partial sums, or partial series
$${S}_{n}:=\sum _{i=1}^{n}{X}_{i}.$$ 
If ${X}_{i}\in \{1,1\}$, then the random walk defined on ${X}_{i}$ is called a simple random walk. A symmetric simple random walk is a simple random walk such that $\mathbf{P}({X}_{i}=1)=1/2$.
The above defines random walks in onedimension. One can easily generalize to define higher dimensional random walks, by requiring the ${X}_{i}$ to be vectorvalued (in ${\mathbb{R}}^{n}$), instead of $\mathbb{R}$.
Remarks.

1.
Intuitively, a random walk can be viewed as movement in space where the length and the direction of each step are random.

2.
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 $\mathrm{\infty}$).

3.
If the random variables ${X}_{i}$ defining the random walk ${w}_{i}$ are integrable with zero mean $\mathrm{E}[{X}_{i}]=0$, ${S}_{i}$ is a martingale^{}.
Title  random walk 

Canonical name  RandomWalk 
Date of creation  20130322 14:59:22 
Last modified on  20130322 14:59:22 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 60G50 
Classification  msc 82B41 
Defines  simple random walk 
Defines  symmetric simple random walk 