I hope that I should describe the original question rather than the forms after some simplifications.
Given two independent $N$-dimensional Gaussian vectors $n_{1}\sim\mathcal{N}(0,\sigma^{2}_{1}\mathrm{I}_{N})$, $n_{2}\sim\mathcal{N}(0,\sigma^{2}_{2}\mathrm{I}_{N})$.
And I hope to calculate the following probability.
\begin{equation} \mathbb{P}(|n_{1}+n_{2}|^{2}\leq|n_{1}|^{2})\quad (3) \end{equation} where $\sigma_{2}<\sigma_{1}$ and $|\cdot|^{2}$ is the $2$-norm.
Intuitively, the above probability tends to $0$ as $N$ tends to $\infty$ since in the high-dimensional space, two independent vectors are almost orthogonal.
And I want to investigate the asymptotic behavior of the above probability when we choose a sufficiently large $N$, i.e., the decreasing speed with $N$ of the above probability. I have also done some numerical experiments to verify that the above probability tends to zero as the increasing of $N$.
And after some calculations I obatin that the above probability equals the following integral. \begin{equation} \mathbb{P}(|n_{1}+n_{2}|^{2}\leq|n_{1}|^{2})=\int_{0}^{\infty}\Phi(-\frac{-\sigma_{2}\sqrt{z}}{2\sigma_{1}})\frac{z^{N/2-1}\exp{-z/2}}{2^{N/2}\Gamma(N/2)}dz \end{equation} where $\Phi(\cdot)$ is the CDF of the standard Gaussian distribution.
Is the result I obatained correct? If not, what should I do to calculate the correct result of (3)?
And there is a link with the previous question. Asymptotic Expansions for the Hypergeometric function for large parameters and bounded z?
Thanks a lot for any advisable suggestions !