Prove if a stochastic process is a white noise

Question

I am trying to show if a stochastic process is a weak and strict white noise. The exercise is as follows:

Let $\{\epsilon\}_{t \in \mathbb{Z}}$ be a strict white noise, $Var(\epsilon_t)=\sigma^2$, $E(\epsilon^3_t)=m_3$ and $E(\epsilon^4_t)=m_4$. Let $\{Y_t\}_{t \in \mathbb{Z}}$ be a process, defined as $Y_t=\epsilon^2_{t-1}\epsilon_t$.

I understand that to show if the process is a weak white noise, one of the conditions that must be met is that $\forall s, t$, $s\neq t$ $Cov(\epsilon_s, \epsilon_t)=0$, and I find myself having problems in obtaining this covariance, given the nonlinearity of the process.

Thank you for your help.

Answer 1

Be careful the resuts that you want to prove is for $t \ne s$. Let assume that $t>s$

$$cov(Y_t,Y_s)=cov(\epsilon^2_{t-1}\epsilon_t,\epsilon^2_{s-1}\epsilon_s)=E[\epsilon^2_{t-1}\epsilon_t\epsilon^2_{s-1}\epsilon_s]-E[\epsilon^2_{t-1}\epsilon_t]E[\epsilon^2_{s-1}\epsilon_s]$$

if $s \ne t-1$ $$E[\epsilon^2_{t-1}\epsilon_t\epsilon^2_{s-1}\epsilon_s]=E[\epsilon^2_{t-1}]E[\epsilon_t]E[\epsilon^2_{s-1}]E[\epsilon_s]$$ and

$$E[\epsilon^2_{t-1}\epsilon_t]E[\epsilon^2_{s-1}\epsilon_s]=E[\epsilon^2_{t-1}]E[\epsilon_t]E[\epsilon^2_{s-1}]E[\epsilon_s]$$ because the $\epsilon_t$ are iid. By definition of a strict white noise, the mean is nil and variance finite. Therefore $cov(Y_t,Y_s)=0$ Same argument holds,if $s=t-1$, you add the fact that the third moment of the white noise is finite.