True/False : Stationary process In stochastic process

[hojoon yang]

Stochastic process problem!

1. If Xn and Yn are independent stationary process, then Vn= Xn / Yn is wide-sense stationary. (T/F)

2. If Xn and Yn are independent wide sense stationary process, then Wn = Xn / Yn is wide sense stationary (T/F)

I solve this problem like this:

1. E[Vn]=E[Xn/Yn], since independent E[Xn]*E[1/Yn] <- using this theorem E[g(x)*f(y)]=E[g(x)]*E[f(y)]
here, I knew it E[Xn]=μx,E[Yn]=μy, clearly not depend on 'n'

But I'm not sure E[1/yn] is not depend on 'n'

Help me please...

 

[andrewkirk]

For the second one: to be wide-sense stationary, all that is required is that E[Yn] is independent of n and that autocorrelations are independent of n.
That leaves us free to make the standard dev anything we want.
Consider a very simple case where Yn can have values $3\pm \left(1+(n \mod 2)\right)$, each with 50% probability.
The mean of Yn is 3, which is independent of n. What is the mean of 1/Yn? Does it depend on n?

For the first one, write E[1/Yn] as an integral and decide whether the integral depends on n, using the fact that Yn is fully stationary, not just wide-sense.