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The Problem

Given a gaussian random variable $\mathbf{x} = (x_1, ..., x_N)^T \sim \mathcal{N}(0, \Sigma)$ what is the probability that $i = \underset{i\in\{1, \dots, N\}}{argmax}\{|x_1|, \dots, |x_N|\}$?

An Example

Lets say that we are dealing with bivariate normal random variables such that $\mathbf{x} = (x_1, x_2) \sim N(0, \Sigma)$. The problem can be stated geometrically as, what is the probability that the norm of the vector $\mathbf{x}$ falls in region 1 vs region 2 of the unit circle.

Hand drawn picture to help illustrate problem

Note that all internal angles are at 45 degree increments which may be useful for a solution to this problem.

While I illustrate this with a bivariate normal random variable I want an answer that holds for arbitrary finite dimensions.

My Attempts So Far

My initial thoughts are to simplify the problem somewhat by using a reparameterization of $\mathbf{x}$ in terms of the eigendecomposition of $\Sigma = U\Lambda U^T = U\Lambda^{1/2}(U\Lambda^{1/2})^T$. $$\mathbf{x} \sim U\Lambda^{1/2}\mathcal{N}(0, I).$$ My thought is that this problem is easy if $\Sigma = I$ as the answer would then be (by symmetry) that $$P\left(i = \underset{i\in\{1, \dots, N\}}{argmax}\{|x_1|, \dots, |x_N|\}\right) = 1/N$$.

If the problem were framed as an expectation, then this reparameterization would allow us to calculate the difficult integral and the factor $U\Lambda^{1/2}$ would come out of the integral as a constant factor leading to the solution.

Am I onto something here?

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