I have been studying a bivariate random process where $X \sim N(0, \sigma_x), Y \sim N(0, \sigma_y)$. It turns out that finding an ellipse that covers proportion p of samples on this process is given by elementary functions: Its semi-axes are $a = \sigma_x \sqrt{-2 \ln(1-p)}$ and $b = \sigma_y \sqrt{-2 \ln(1-p)}$.
However, the analogue in one dimension is not so elegant: The interval that covers proportion p of just one of the variables (say X) is given by $\sigma_x \sqrt{2} \text{erf}^{-1}(p)$.
My naive intuition is that this p-interval in one dimension should be the limit of the p-ellipse in two dimensions as $\sigma_y \rightarrow 0$. But given the formulas above this does not appear to be the case.
Why is the one dimensional covering interval not the limit of the two-dimensional covering ellipse? Or is it, and it can be shown that the formulas for the ellipse axes collapse to the (non-elementary) inverse error function in the limit?