Random Walk

A Random Walk, also called random walk or random walk is a mathematical model for a movement in which the individual steps are carried out at random. It is a stochastic process in discrete time with independent and identically distributed increments. Random walk models are suitable for non-deterministic time series, as they are used for example in financial mathematics for modeling stock prices (see random walk theory). With their help, the probability distributions of measured values ​​of physical quantities can be understood. The term goes back to Karl Pearson's essay The Problem of the Random Walk from the year 1905.

Definition

Let be a sequence of independent random variables with values ​​in which all have the same distribution. Then by means of

Defined stochastic process is a random walk in or a d -dimensional random walk. It is often chosen. A random walk is thus a discrete process with independent and stationary increments.

One-dimensional case

The simple one-dimensional random walk is a basic tutorial, which extended to multiple dimensions and can be generalized; but he has already itself a number of concrete applications. In the one-dimensional random walk the steps form a Bernoulli process, that is a sequence of independent Bernoulli trials.

A popular illustration is as follows ( see also Drunkard 's Walk ): A disoriented pedestrians running in an alley with a probability a step forward and one step back with a probability. His random position after steps is referred to as, without limitation, was his starting position at. What is the probability that he is right in the nth step in the place? Answer: The pedestrian has made ​​a total of steps, of which steps forward and steps back. His position is so steps and the probability is

As the number of forward steps follows a binomial distribution.

Often one is interested in specifically for the undirected or symmetric random walk. The probability distribution of the distance traveled is symmetrical about, and the expectation value. The progress of the pedestrian can then only by the mean square distance from the starting point, ie, by the variance of the binomial distribution to describe. This is an important result, with the unique characteristic of diffusion processes and Brownian molecular motion is found again: The mean square of the distance of a diffusing particle from its starting point increases proportionally to time.

A first generalization is that at each step a random step length is allowed. The figure shows an example of five simulations for steps with a step length that is uniformly distributed in the interval. In this case, the standard deviation is, for each step. The standard deviation of such a random walk with steps is then units. She is shown as a red line for positive and negative distances. To cover this distance the pedestrian will move on average. The relative deviation goes to zero, the absolute deviation, however, grows unbounded.