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I need to calculate or at least estimate the expected value of this term

$$ \mathbb{E} \left[\frac{1}{(L-K \cdot\left \| \mathcal{N}(0,\Sigma) \right \|_2)^2}\right] $$

where $L,K\in \mathbb{R}>0$ are constants and $\left \| \mathcal{N}(0,\Sigma) \right \|_2$ stands for the 2-Norm of a zero mean multivariate normal distribution. Since $\mathbb{E}[1/X] \neq \mathbb{E}[1]/\mathbb{E}[X]$ for almost every case, I'm stuck on how to approach this problem.

user67080
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    https://stats.stackexchange.com/questions/299722 might be relevant. It immediately resolves the question when $L$ and $K$ are of the same sign, for then the expectation must be infinite. As as "for almost every case" goes, you can refine that to state "except when $X$ is almost surely a nonzero constant." – whuber Feb 14 '22 at 14:26
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    Since you have specified $L$ and $K$ are both positive, the link I gave fully answers your question once you observe that the denominator is continuous at zero with a positive density there. – whuber Feb 14 '22 at 14:59
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    Thank you for your reply and the link. Helped me a lot ! – user67080 Feb 14 '22 at 15:38

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