# 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. 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 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. 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 RandomWalk 2013-03-22 14:59:22 2013-03-22 14:59:22 CWoo (3771) CWoo (3771) 7 CWoo (3771) Definition msc 60G50 msc 82B41 simple random walk symmetric simple random walk