Definition. Let be a probability space and a discrete-time stochastic process defined on , such that the are iid real-valued random variables, and , the set of natural numbers. The random walk defined on is the sequence of partial sums, or partial series
If , then the random walk defined on is called a simple random walk. A symmetric simple random walk is a simple random walk such that .
The above defines random walks in one-dimension. One can easily generalize to define higher dimensional random walks, by requiring the to be vector-valued (in ), instead of .
Intuitively, a random walk can be viewed as movement in space where the length and the direction of each step are random.
It can be shown that, the limiting case of a random walk is a Brownian motion (with some conditions imposed on the 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 ).
If the random variables defining the random walk are integrable with zero mean , is a martingale.
|Date of creation||2013-03-22 14:59:22|
|Last modified on||2013-03-22 14:59:22|
|Last modified by||CWoo (3771)|
|Defines||simple random walk|
|Defines||symmetric simple random walk|