random walk

Definition. Let $(\Omega,\mathcal{F},\mathbf{P})$ be a probability space and $\{X_{i}\}$ 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 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 $X_{i}\in\{-1,1\}$ , 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 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 $\mathbb{R}$ .

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 $\infty$ ).
3.

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.

Title	random walk
Canonical name	RandomWalk
Date of creation	2013-03-22 14:59:22
Last modified on	2013-03-22 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