Let Z 0 , Z 1 , Z 2 ,... be a sequence of independent identically distributed mean zero normal random variables with variance one. Define   (1) Show that the convergence in (6.9) is absolute and...

Let Z₀, Z₁, Z₂,... be a sequence of independent identically distributed mean zero normal random variables with variance one. Define

(1) Show that the convergence in (6.9) is absolute and uniform over

(2) Show that X_t
is a Gaussian process.

(3) If W_t
is a Brownian motion and

show that X and Y have the same finite-dimensional distributions. Show that X and Y have the same law when viewed as random variables taking values in C[0, 1]. (The process X is sometimes known as integrated Brownian motion.)

(4) Find Cov (X_s, X_t).

Chapter 6