# Prove if a stochastic process is a white noise

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.

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.
• By definition of a white noise, the expectation of $e_t$ is zero, the rest you wrote it on your post Commented Dec 5, 2019 at 5:12
• I meant how you know this is true: $E(\epsilon^2_{t-1}\epsilon_t\epsilon^2_{s-1}\epsilon_s)=E(\epsilon^2_{t-1})E(\epsilon_t)E(\epsilon^2_{s-1})E(\epsilon_s)$ Commented Dec 5, 2019 at 5:13
• That is the definition I have of a strict/strong white noise, the $\epsilon_t$ are iid, so they are independent. Commented Dec 5, 2019 at 5:15