Overview
Brownian motion is shown to exist by providing a construction. That is, it is the limit of a sequence of functions that are constructed below.
Brownian Motion Construction
Define a sequence of random variables {% X_i %}, such that each takes on a value of 1 or -1 with probability 0.5 each. Then define
{% M_j = \sum_{i=1}^j X_i %}
{% W_{n,t} = \frac{1}{\sqrt{n}} M_{n,t} %}
Theorem
The distribution of {% W_{n,t} %} converges to mean zero, variance {% t %} normal distribution as {% n \rightarrow \infty %}
Visualization
Shows a Visualization of the above construction. Moving the slider increases the value of {% n %}. Brownian motion is the limit of this process as {% n \rightarrow \infty %}