|
|
|
Viewing Version
2
of
'random walk'
|
[ view 'random walk'
|
back to history
]
| Title of object: |
random walk |
| Canonical Name: |
RandomWalk |
| Type: |
Definition |
| Created on: |
2005-01-31 20:14:39 |
| Modified on: |
2005-01-31 21:41:36 |
| Classification: |
msc:60G50, msc:82B41 |
| Defines: |
simple random walk, symmetric simple random walk |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb,amscd}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
\textbf{Definition}. Let $(\Omega,\mathcal{F},\mathbf{P})$ be a
probability space and $\lbrace x_i \rbrace$ a discrete-time
stochastic process defined on $(\Omega,\mathcal{F},\mathbf{P})$,
such that the $x_i$ are iid real-valued random variables, and
$i\in\mathbb{N}$, the set of natural numbers. The \emph{random
walk} defined on $x_i$ is the sequence of partial sums, or partial
series $$w_n\colon=\sum_{i=1}^{n}x_i.$$ If $x_i\in\lbrace -1,1
\rbrace$, then the random walk defined on $x_i$ is called a
\emph{simple random walk}. A \emph{symmetric simple random walk} is
a simple random walk such that $\mathbf{P}(x_i=1)=1/2$.
The above defines random walks in one-dimension. One can easily
generalize to define higher dimensional random walks, by requiring
the $x_i$ to be vector-valued (in $\mathbb{R}^n$), instead of
$\mathbb{R}$.
\textbf{Remarks}.
\begin{enumerate}
\item Intuitively, a random walk can be viewed as movement in space
where the length and the direction of each step are random.
\item It can be shown that, the limiting case of a random walk is a
Brownian motion (with some conditions imposed on the $x_i$ 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 $\infty$).
\item If the random variables $x_i$ defining the random walk $w_i$
are integrable with zero mean $\operatorname{E}(x_i)=0$, $w_i$ is a
martingale.
\end{enumerate} |
|
|
|
|
|
