Analytic asymptotic null distribution of the quotient-affine distance

Derive an analytic asymptotic null distribution for the quotient-affine geodesic distance between sample correlation matrices.

Background

The quotient-affine geodesic distance provides an intrinsic global measure of separation between full-rank correlation matrices. Before the paper's local asymptotic argument, an analytic null distribution for this nonlinear distance was not available to the authors.

The paper subsequently derives a Gaussian local asymptotic result by relating the squared geodesic distance to Jennrich's quadratic statistic, showing convergence after effective-sample-size scaling to a chi-squared distribution. The explicitly stated unresolved issue concerns obtaining an analytic asymptotic null distribution for the distance itself, beyond its local connection to Jennrich's statistic.