Let $\epsilon \sim N\left ( 0,I_{k} \right )$, consider the following function of $\epsilon$,

$y=\left ( A+B\epsilon \epsilon {}'B{}' \right )^{^{\frac{1}{2}}}\epsilon $,

where $A$ is a positive semi-definite symmetric $k\times k$ matrix, $B$ is a $k\times k$ matrix. I want to find the distribution or the approximate distribution of $y$, or some moments of $y$, such as $E\left ( yy{}' \right )$, $E\left ( yy{}'\otimes yy{}' \right )$.