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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!?

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  • $\begingroup$ Can you write an expression for the energy of the random signal? There were some comments earlier on this question but they seem to have been deleted. $\endgroup$ – Dilip Sarwate Nov 19 '15 at 15:18
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I'm not totally sure I understand your question: could you give a concrete example? For instance, what's the randomness? Anyway, here are some relations between the quantities you're interested in.

The energy of the signal is $$\mid \mid Y \mid \mid_2^2 = \sum_{i=0}^n y_i^2$$

The Auto-correllation matrix $R$ is formed as $YY^T$ (outer product between the signal and it's transpose).

Since this matrix is symmetric, it's matrix 2-norm is the spectral radius: $$||YY^T||_2^2 = \mathrm{max}|\lambda_j|$$

Another way to write the 2-norm would be as $$||YY^T||_F^2 = Tr(YY^T)$$ It's fairly stragtforward to prove that $$||* ||_F^2 \geq ||*||_2^2$$ for any matrix, and they co-incide when $YY^T$ has rank 1. It's also true that $$|| * ||_F^2 \leq r|| *|| _2^2 $$

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