Sample canonical correlation coefficients of high-dimensional random vectors: Local law and Tracy–Widom limit

Abstract

Consider two random vectors [Formula: see text] and [Formula: see text], where the entries of [Formula: see text] and [Formula: see text] are i.i.d. random variables with mean zero and variance one, and [Formula: see text] and [Formula: see text] are respectively, [Formula: see text] and [Formula: see text] deterministic population covariance matrices. With [Formula: see text] independent samples of [Formula: see text], we study the sample correlation between these two vectors using canonical correlation analysis. Under the high-dimensional setting with [Formula: see text] and [Formula: see text] as [Formula: see text], we prove that the largest sample canonical correlation coefficient converges to the Tracy–Widom distribution as long as we have [Formula: see text] and [Formula: see text], which we believe to be a sharp moment condition. This extends the result in [19], which established the Tracy–Widom limit under the assumption that all moments exist for the entries of [Formula: see text] and [Formula: see text]. Our proof is based on a new linearization method, which reduces the problem to the study of a [Formula: see text] random matrix [Formula: see text]. In particular, we shall prove an optimal local law on its inverse [Formula: see text], called resolvent. This local law is the main tool for both the proof of the Tracy–Widom law in this paper, and the study in [26, 27] on the canonical correlation coefficients of high-dimensional random vectors with finite rank correlations.