Title: Tracy-Widom at each edge of real covariance and MANOVA estimators,
Abstract: We study the sample covariance matrix for real-valued data with general
population covariance, as well as MANOVA-type covariance estimators in variance
components models under null hypotheses of global sphericity. In the limit as
matrix dimensions increase proportionally, the asymptotic spectra of such
estimators may have multiple disjoint intervals of support, possibly
intersecting the negative half line. We show that the distribution of the
extremal eigenvalue at each regular edge of the support has a GOE Tracy-Widom
limit. Our proof extends a comparison argument of Ji Oon Lee and Kevin
Schnelli, replacing a continuous Green function flow by a discrete Lindeberg
swapping scheme.