# Category Archives: Construction of Brownian Motion

## Levy’s construction of Brownian Motion

Let $$\{ \xi_k^{(n)} : n =0,1,\dots ; k =1,\dots,2^n\}$$ be a collection of independent Gaussian random variables with  $$\xi_k^{(n)}$$ having mean zero and variance $$2^{-n}$$. Define the random variable $$\eta_k^{(n)}$$ recursively by

$\eta_1^{(0)} = Z \qquad\text{with}\quad Z\sim N(0,1) \quad\text{and independent of the $$\xi$$’s}$

$\eta_{2k}^{(n+1)} = \frac12\eta_{k}^{(n)} -\frac12 \xi_{k}^{(n)}$

$\eta_{2k-1}^{(n+1)} = \frac12\eta_{k}^{(n)} +\frac12 \xi_{k}^{(n)}$

For any time $$t \in [0,1]$$ of the form $$t=k 2^{-n}$$ define

$W^{(n)}_t = \sum_{j=1}^k \eta_{j}^{(n)}$

For $$t \in [0,1]$$ not of this form we connect the two nearest defined points with a line.

1. Follow given steps to show that for fixed $$n$$, $$W^{(n)}_t$$ is random walk on $$\mathbf R$$ with Gaussian steps.
1. Show $$\mathbf E \eta_{k}^{(n)} = 0$$ and  $$\mathbf E \big[ (\eta_{k}^{(n)})^2\big] = 2^{-n}$$
2. Argue that $$\eta_{k}^{(n)}$$ is Gaussian and that for any fixed $$n$$,
$\{ \eta_{k}^{(n)} : k=1,\dots, 2^n\}$
are a collection of mutually independent random variables. (To show independence show that they are mean zero Gaussians  with correlation $$\mathbf E [\eta_{k}^{(n)}\eta_{j}^{(n)}]=0$$ when $$j\neq k$$.)
2. To understand the relationship between $$W^{(n)}$$ and $$W^{(n+1)}$$, simulate a collection of random $$\xi_k^{(n)}$$ and plot $W^{(0)}, W^{(1)}, W^{(2)}, W^{(3)}, W^{(4)}$
over the time interval $$[0,1]$$. Notice that at $$n$$ increases the functions seem to converge. Try a few different realizations to get a feeling for how the limiting function might look.