Relation between Covariance matrix & Energy of a random signal
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 $$|| * ||_F^2 \leq r|| *|| _2^2 $$