# pr.probability – Average value of \$frac{x’A^2x}{x’A^3x}\$ over surface of \$n\$-dimensional sphere

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. Define $$z_n$$ as follows
$$z_n=E_{xsim mathcal{N}left(0, I_nright)}left(frac{x^T A^2 x}{x^T A^3 x}right)$$

Is it possible to prove or disprove the following?

$$lim_{nto infty} z_n = 2$$

This is a crosspost from math.SE where several people provided altnernative characterizations of $$z_n$$ but which don’t quite settle the question.

Motivation: $$z_n$$ is the expected value of learning rate which maximizes loss decrease for a gradient descent step on a quadratic $$A$$ and random starting point. If the limit is 2, this would be a nice mathematical illustration behind the heuristic used in practice, “in high dimensions — set learning rate as high as possible”