Random Walk Examples

There is a drunk man trying to make his way home. He has no sense of direction and has decided that he should only move right or left. He will not move in a combination of these directions, such as North East or South West. As he stumbles home the probability of him turning in any direction are all equal. This event is a stochastic process formed by successive summation of independent, identically distributed random variables (Alm 2). In other words, this scenario is a common example used to describe a random walk. If the man were to decide that he should also move forward or backward, this situation would be an example of a 2D random walk. As the directions, steps, and rules for the drunkard vary further, the random walk becomes more complex

There are many types of random walks categorized by the rules the movement of the random walk follows. Multiple features of a random walk are used to form a full description of it, such as the simple symmetric random walk. Features include, but are not limited to, simple, one-dimensional, higher-dimensional, finite or infinite, symmetric, and Gaussian random

Simply put, a random walk is symmetric when the probability of each step is equal. For example, the number line example used previously is symmetrical, as the probabilities of the movement to neighbouring points are both fifty percent. When a random walk is not symmetric it is bias.

Gaussian random walks, or random walks with Gaussian steps, are random walks that have step sizes that vary according to a normal distribution (Chang 787). While Xi=1in a simple random walk, Xiis distributed normally for a Gaussian random walk. This characteristic means that the values are taken in a symmetrical fashion and are positioned around the mean. The probability of a value places it on the curve, closer or farther from the mean. This description forms a graph known as the bell curve. Overall, Gaussian random walks have varying steps.

To summarize, random walks are described as simple, one-dimensional, higher-dimensional, finite or infinite, symmetric, and Gaussian random walks. Simple random walks have a step size of positive or negative one and, in one dimension, have equal probability for the direction of the step. One dimensional random walks have two neighbouring points, for example left or right, two-dimentional have four, and three-dimentional have six. Finite random walks are noncontinuous and have an end, while infinite ones do not. Symmetric random walks have equal probabilities for each step, and Gaussian