Relaxation of cross-sectional independence and common specific variance assumptions

Determine whether the cross-sectional independence assumption for the specific returns in Assumption 2 can be relaxed to allow weak dependence and whether the common cross-sectional specific-variance assumption across dates in Assumption 3 can also be relaxed.

Background

The paper analyzes principal-component estimation error in a high-dimension, low-sample-size factor model. Its asymptotic results assume that specific-return entries are mutually independent conditional on the factor path and that their cross-sectional average variance converges to the same limit for every observation date.

The authors explicitly conjecture that both assumptions may be weakened: conditional cross-sectional independence might be replaced by weak dependence, and the requirement that the cross-sectional specific variance have a common limit across dates might be removed. No proof or extension is provided in the paper.