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For large $k$ the matrix $\epsilon\epsilon^\top\rightarrow \mathbb{E}[\epsilon\epsilon^\top]=I_k$, with fluctuations that vanish as $1/\sqrt k$, so $y\rightarrow (A+BB^\top)^{1/2}\epsilon$ and the elements of $y$ are Gaussian with zero mean and covariance matrix $C=A+BB^\top$.