Second Order Stationary Process

[cdplayersony]

Homework Statement

Let X(t), -inf< t < inf, be a second order process. Show that it is a second order stationary process if and only if mu(t) sub x is independent of t and r(s,r) sub x depends only on the difference between s and t.

Show that is a second order stationary process if and only if EX(s) and EX(s)X(s+t) are both independent of s.

Set Y(t)=X(t+1)-X(t), -inf< t <inf. Show that the Y(t) process is a second order stationary process having zero means and covariance function: r(t) sub y =2*r(t) sub x -r(t-1) sub x -r(t+1) sub x

Homework Equations

mu(t) sub x is the mean function =EX(t)
r(s,t) sub x is the covariance function EX(s)X(t)-EX(s)EX(t)

The Attempt at a Solution

I know that if r(s,t) depends only on the difference between s and t r(s,t)=r(0, t-s) where r(t)=r(0,t) and this is the auto-covariance function of the process, but I just can't seem to get these relations proven. Thanks for the help!

 

[Stephen Tashi]

The definition of "second order stationary process" should certainly be included under "Relevant Equations" (even though, it is technically not simply an equation). Those of us who don't remember it might be able to help if we are reminded.