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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
 .
Remarks.
- 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
![$ \operatorname{E}[X_i]=0$](http://images.planetmath.org:8080/cache/objects/6694/l2h/img18.png "$ \operatorname{E}[X_i]=0$") ,  is a martingale.
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"random walk" is owned by CWoo. [ full author list (2) ]
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(view preamble)
| Also defines: |
simple random walk, symmetric simple random walk |
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Cross-references: martingale, number, walk, mean, Brownian motion, length, series, partial sums, sequence, natural numbers, random variables, iid, stochastic process, probability space
There are 4 references to this entry.
This is version 4 of random walk, born on 2005-01-31, modified 2006-10-22.
Object id is 6694, canonical name is RandomWalk.
Accessed 7708 times total.
Classification:
| AMS MSC: | 60G50 (Probability theory and stochastic processes :: Stochastic processes :: Sums of independent random variables; random walks) | | | 82B41 (Statistical mechanics, structure of matter :: Equilibrium statistical mechanics :: Random walks, random surfaces, lattice animals, etc.) |
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Pending Errata and Addenda
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