Let's say I have the below random signal: $ Y[n] = [y(n), y(n-1), y(n-2), \ldots, y(1)] $
I have two random variables now: The first one $X_1 $ which express the maximum eigenvalue of the covariance matrix of $Y$. The second one $X_2$ which express the energy of the random signal.
Now my question is: Are the two random variables independent or dependent, when whether signal samples $y(n),y(n-1)\ldots$ are IID or correlated with each other?
By intuition the two random variables should be correlated!? Isn't it right!?