A Functional CLT for Partial Traces of Random Matrices

Abstract

AbstractIn this paper, we show a functional central limit theorem for the sum of the first $$\lfloor t n \rfloor $$ ⌊ t n ⌋ diagonal elements of f(Z) as a function in t, for Z a random real symmetric or complex Hermitian $$n\times n$$ n × n matrix. The result holds for orthogonal or unitarily invariant distributions of Z, in the cases when the linear eigenvalue statistic $${\text {tr}}f(Z)$$ tr f ( Z ) satisfies a central limit theorem (CLT). The limit process interpolates between the fluctuations of individual matrix elements as $$f(Z)_{1,1}$$ f ( Z ) 1 , 1 and of the linear eigenvalue statistic. It can also be seen as a functional CLT for processes of randomly weighted measures.