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!