Prove a time series to be NOT identically independent distributed

I am trying to prove that this time series (given that $$X_{t}$$ and $$M_{t}$$ are iid and independent of each other) $$Y_{t} = X_{t}(1-X_{t-1})M_{t}$$ is not i.i.d, so my understanding is that I need to prove that this series has dependent realizations. Am I correct by making this statement?

We know that if two random variables are independent, then their correlation or covariance is 0.

Can I use this strategy by saying that if their correlation or covariance is equal something else which is not 0, then they are dependent?

For example, can I try to show that if $$Cov(Y_{t},Y_{t+1}) \ne 0$$, then $$\{Y_{t}\}$$ is not iid?

If not, any tip would be great.

Thank you!

• One has to be careful in making such a statement. If $X_t \equiv 0$ or $X_t \equiv 1$ or $M_t \equiv 0$ then $(Y_t)$ is i.i.d.. – Kavi Rama Murthy Feb 19 at 5:35
• But your general plan, of using correlation to imply dependence, is good. – kimchi lover Feb 19 at 11:38
• Thanks folks! I appreciate your inputs! – Adam Ralphus Feb 19 at 15:48