In this and the next exercise we give the Haar function construction of Brownian motion. Let set It is a well-known and easily proved result from analysis (see, e.g., Bass (1995), Section I.2)...

In this and the next exercise we give the Haar function construction of Brownian motion. Let

set

It is a well-known and easily proved result from analysis (see, e.g., Bass (1995), Section I.2) that the collection

is a complete orthonormal system for L₂[0, 1]. For each i, j, define

for each i and j, let Y_ij
be independent mean zero normal random variables with variance one, and let.