Let be a complete orthonormal system for L2 [0, 1]. Show that the sum (6.8) converges in L2 and give the details of the proof that the resulting process W is a mean zero Gaussian process with Let ...

Let

be a complete orthonormal system for L₂
[0, 1]. Show that the sum (6.8) converges in L₂
and give the details of the proof that the resulting process W is a mean zero Gaussian process with

Let

be the dyadic rationals. Suppose the collection of random variables

is jointly normal, each V_t
has mean zero, and Cov (V_s,V_t) = s∧t. (1) Prove that the paths of V are uniformly continuous over t ∈ D.

(2) If we define

prove that W is a Brownian motion.