TITLE: Average value of $\frac{x'A^2x}{x'A^3x}$ over surface of $n$-dimensional sphere
QUESTION [12 upvotes]: Suppose $A$ is a diagonal matrix with eigenvalues $1,\frac{1}{2},\frac{1}{3},\ldots,\frac{1}{n}$ and $x$ is drawn from standard Gaussian in $n$ dimensions. In numerical simulations, the following quantity seems to converge to $2$ as $n\rightarrow \infty$
$$z_n=E_{x\sim \mathcal{N}\left(0, I_n\right)}\left[\frac{x^T A^2 x}{x^T A^3 x}\right]$$
Can this be proven or disproven?
$z_n$ can also be written as the following sum
$$z_n=\sum_{i=1}^n i E_{y\sim \mathcal{N}\left(0,A^3\right)}\left[\frac{y_i^2}{\|y\|^2}\right]$$
This quantity can be viewed as the average ratio of quadratic forms $A^2$ and $A^3$ on the surface of $n$-dimensional sphere.

Here's what the distribution looks like for a few values of $n$, means are tending towards $2$

REPLY [2 votes]: I think this is covered, but not very accessible, in this paper from 1953 by Gurland on "Distribution of Quadratic Forms and Ratios of Quadratic Forms". It considers ratios for positive symmetric $Q$ and $P$ for:
$$\frac{x^TQx}{x^TPx}$$
https://projecteuclid.org/journals/annals-of-mathematical-statistics/volume-24/issue-3/Distribution-of-Quadratic-Forms-and-Ratios-of-Quadratic-Forms/10.1214/aoms/1177728981.full
Also this reference may be of interest:
https://www.researchgate.net/publication/224817325_Quadratic_Forms_in_Random_Variables_Theory_and_Applications
Edit:
(I didn't know of that other posting. My references above were redundant.)
I think @DinosaurEgg is right. I verified that integral by the one given in this reference (for integer exponents):
https://www-2.rotman.utoronto.ca/~kan/papers/mupq.pdf
I also numerically integrated up to n=200000 and got the value 1.99217
It seems to stabilize to a value below 2.0 so most likely the value is not 2.0.
I also provided some graphs below to build some level of confidence but certainly no proof.
Convergence (expectation vs. n):

Integrand (s<=1000, n=200000):

Log of Integrand (s<=1000, n=200000):