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.



Comments are closed.