Determining stationary and mean-ergodicity

[rsingh628]

Hi all, I need to determine whether the random process below is strict-sense stationary (SSS), whether it is wide-sense stationary (WSS), and whether it is ergodic in the mean. All help is much appreciated.

[imath] X_k = e^-|k - D| [/imath] where [imath] D [/imath] ~ [imath] Poisson(3) [/imath] (exp is raised to the -|k - D| )

By definition to be WSS, the mean and autocorrelation function need to be independent of time k, and to be SSS, all statistics need to be independent of time, i.e. invariant to time delays. For mean-ergodic, the ensemble and time averages need to be equal. My issue is setting up the equations and justifying.

Approach:
Since the Poisson RV includes a time duration in its PDF, then [imath] X_k [/imath] would depend on time, so it is not WSS.
To test for SSS, I simply replace [imath] k [/imath] with [imath] k - d [/imath] , so [imath] X_(k-d) = e^-|k-d - D| [/imath], but since it is not WSS, then it would still dependent on time, so it is not SSS
I'm not sure how to approach proving if the ensemble and time averages are the same or not. Any help appreciated.

 

[blamocur]

Since the Poisson RV includes a time duration in its PDF, then Xk X_k Xk would depend on time, so it is not WSS.

I don't know enough about stochastic processes, so take with a grain of salt, but: I am not sure I agree with the quoted statement. What if you had, for example, [imath]X_k = e^{-D}[/imath] ?

But I suspect that [imath]E(X_k)[/imath] does depend on [imath]k[/imath] (and I'd try showing that if it were my homework), which makes it non-WSS (and, most likely, not SSS).