I have been reading Peebles Probability, Random Variables, and Random Signal Principles and it claims that second-order stationarity is sufficient to guarantee:

  1. $E[X(t)]$ is a constant
  2. $R_{XX}(t1,t2) = E[X(t2)X(t2)]$ depends only on the difference $t1-t2$ of the two arguments and not on the individual arguments $t1$ and $t2$.

However it says that second-order stationarity is often more restrictive than necessary and hence the concept of a process that just complies with this conditions, named wide-sense stationary process, was created.

A second-order stationary process is defined as having a second order density function that is only dependent of time difference.

I would like to know:

  1. What is meant by "more restrictive than necessary"? Is it just having to prove that second order density function is dependent only of time difference? What other aspects does a second-order stationary process has that a wide-sense stationary process doesn't?
  2. If a second-order stationary process is also first-order stationary. What would a wide-sense stationary process require to be first-order stationary? A first-order density function that does not change with a shift in time?
  3. How high must the N in Nth-order stationary process be to be considered a Strict-Sense Stationary process? Infinite?


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