Weak convergence in circulant matrices (Q2576812)

A circulant matrix is a real square matrix such that each row is obtained by shifting the previous row cyclically. The authors consider a probability measure on the Borel sets of \(d\times d\) circulant matrices, and consider the problem of weak convergence of the convolution sequence \((\mu^n)_{n\geq1}\) on the Borel sets of the closed multiplicative semigroup generated by the support of \(\mu\). The authors characterize, in the \(3\times3\) and \(4\times4\) cases, the kernel \(K\) of \(S\) under the assumption that \((\mu^n)\) is tight. The same kind of results are obtained for certain Toeplitz matrices.