random walk

Definition. Let (Ω,,𝐏) be a probability spaceMathworldPlanetmath and {Xi} a discrete-time stochastic process defined on (Ω,,𝐏), such that the Xi are iid real-valued random variablesMathworldPlanetmath, and i, the set of natural numbers. The random walkMathworldPlanetmath defined on Xi is the sequence of partial sums, or partial series


If Xi{-1,1}, then the random walk defined on Xi is called a simple random walk. A symmetric simple random walk is a simple random walk such that 𝐏(Xi=1)=1/2.

The above defines random walks in one-dimension. One can easily generalize to define higher dimensional random walks, by requiring the Xi to be vector-valued (in n), instead of .


  1. 1.

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

  2. 2.

    It can be shown that, the limiting case of a random walk is a Brownian motionMathworldPlanetmath (with some conditions imposed on the Xi 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 ).

  3. 3.

    If the random variables Xi defining the random walk wi are integrable with zero mean E[Xi]=0, Si is a martingaleMathworldPlanetmath.

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