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A functional CLT for partial traces of random matrices. (English) Zbl 1466.60011

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

MSC:

60B20 Random matrices (probabilistic aspects)
60F17 Functional limit theorems; invariance principles
60F05 Central limit and other weak theorems
15B52 Random matrices (algebraic aspects)
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