Question: Prove eq. 2.49-
Separating the complex process into its real and imaginary parts, we get two independent Brownian motions. Now that Z(t) has been shown to satisfy all the requirements of the definition of a Brownian motion, but the continuity property, it remains to show that almost all its paths are continuous. To show that the paths of Z(t) are continuous, the almost sure (in ?˜) uniform convergence of the series (2.48) has to be demonstrated. Once this is done, the space of realizations of the infinite sequence Zk can be identified with the space ? of continuous functions through the one-to-one correspondence (2.48). Thus, we write Z(t, ω) to denote any realization of the path. For any ω denote
