Prove Weakly Stationary Process w/$E(X^2_t ) < ∞$

[nacho-man]

Not 100% sure if this is the right board.

My question is to
Show that a strictly stationary process with $E(X^2_t ) < ∞$ is weakly stationary.

So weakly stationary implies two things:

- the mean value function $u_t$ does not depend on time $t$
and
- the autocovariance $\gamma_x(t+h,t)$ is independent of $t$ for each $h$

The first point is fairly intuitive, but just to clarify, the second point is saying that the ACF does not depend on the time $t$ at any point when finding the autocovariance between two different points in time for any given $h$ ?Anyway, continuing on to the question,

the solutions say:

$E[X_t]$ is independent of $t$ since the distribution of ${X_t}$ is independent of $t$ and $E[X_t]$ exists.
So that proves the first point, but isn't this just an obvious re-iteration of the first definition of a weakly stationary set?
How does the 2nd moment being finite allow us to conclude that the distribution of ${X_t}$ is independent of $t$?

The 2nd point of the solution says: the joint distribution of $X_t , X_{t+h}$ is independent of t, hence ${X_t}$ is weakly stationary.

Are these both valid solutions? It seems as if just the definition of a weakly stationary set has been provided. I feel a little confused as to why these conclusions cannot be made if $E^2[X] \nless \infty$

Any help with understanding this is very appreciated.

 

[tarantella]

Hello,

The assumption of finiteness of $\mathbb E(X_t^2)$ is done because otherwise the quantities $\mathbb E(X_tX_{t+h})$ involved in the definition of weak stationarity do not necessarily make sense.

The solution is valid.