Abstract
Combining Stein's method with heat kernel techniques, we show that the trace of the jth power of an element of U(n,C), USp(n,C), or SO(n,R) has a normal limit with error term C j/n, with C an absolute constant. In contrast to previous works, here j may be growing with n. The technique might prove useful in the study of the value distribution of approximate eigenfunctions of Laplacians.
Highlights
There is a large literature on the traces of powers of random elements of compact
Using the method of moments, they show that if M is random from the Haar measure of the unitary group U (n, C), and Z = X + iY is a standard complex normal with X and Y
The current paper studies the distribution of T r(M j ) using Stein’s method and heat kernel techniques
Summary
There is a large literature on the traces of powers of random elements of compact. Lie groups. Stein’s method, heat kernel, and traces of powers an iterative version of “Stein’s method” to show that for j fixed, T r(M j) on O(n, R) is asymptotically normal with error O(n−r) for any fixed r. The current paper studies the distribution of T r(M j ) using Stein’s method and heat kernel techniques. We note two differences with her work She uses geodesic flows and Liouville measure instead of heat kernels. Her infinitesimal version of Stein’s method [23],. Fulman and Röllin [13] modify the technique in the current paper to prove a central limit theorem for the trace of AO, where A is a fixed n × n real matrix, and O is from the Haar measure of the orthogonal group O(n, R).
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