Definition:Random Walk/One Dimension

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Definition

Let $\sequence {X_n}_{n \mathop \ge 0}$ be a Markov chain whose state space is the set of integers $\Z$.

Let $\sequence {X_n}$ be such that $X_{n + 1}$ is an element of the set $\set {X_n + 1, X_n, X_n - 1}$.


Then $\sequence {X_n}$ is a one-dimensional random walk.


Examples

Gambling Game

Consider the gambling game in which a player with initial capital $k$ wins $1$ unit with probability $p$ and loses $1$ unit with probability $1 - p$.

This is a one-dimensional random walk with absorbing state $k = 0$.


Also known as

A one-dimensional random walk is also known as a simple random walk.

It can also be seen unhyphenated: one dimensional random walk.


Also see

  • Results about one-dimensional random walks can be found here.


Sources