properties of strictly stationary martingale difference sequence

#1
hey guys,

I couldnt find a similar thread, so I decided to post my problem...

Im in a time series setting: lets describe a sequence [math]\epsilon_t[/math], which got the properties
  • strictly stationary
  • martingale difference sequence with a fixed variance

based on this information, I know the sequence wont correlate with any function of its lags (due mds), is White Noise (due fixed variance) and joint density wont depend on time (strictly stationary).

The task is, to show that [math]Y_t = (1+\epsilon_{t-1}^{2})^{0.5}\epsilon_t[/math] is strictly stationary white noise and also a mds.

showing that [math]y_t[/math] is a mds is no problem. thus, [math]y_t[/math] wont exhibit any correlation with its past and [math]E(y_t)=0[/math]. Next thing would be to show that variance is independent of time to establish that [math]y_t[/math] is also WN.

[math]Var(y_t)=E(y_t^2)=E(\epsilon_t^2)+E(\epsilon_{t-1}^2\epsilon_t^2)[/math]

the first term is obvious, but the secound term puzzles me.
as far as I understood the concepts [math]E(\epsilon_{t-1}^2\epsilon_t^2)[/math] will be invariant to time due the strictly stationarity property of [math]\epsilon_t[/math] (functions of epsilon will also be strictly stationary...)?

besides: Im not allowed to rewrite [math]E(\epsilon_{t-1}^2\epsilon_t^2)[/math] into [math]E(\epsilon_{t-1}^2)E(\epsilon_t^2)[/math] since it could potentially be correlated (these are higher order moments of epsilon), right?

okay, so [math]y_t[/math] at this point is a mds and white noise, thus at least weakly stationary.

I somehow must show that also the joint density of [math]y_t[/math] is time invariant...how? Can I just argue, since [math]y_t=f(\epsilon_t,\epsilon_{t-1})[/math] the whole **** will also be strictly stationary?

sorry for the long post, but Im very confused right now. hope someone is willing to help me out :)

thanks in advance!