Limiting spectral distribution of a class of Hankel type random matrices

We consider an indexed class of real symmetric random matrices which generalize the symmetric Hankel and Reverse Circulant matrices. We show that the limiting spectral distribution of these matrices exists almost surely and the limit is continuous in the index. We also study other properties of the limit and, in particular, explicitly characterize it for a certain subclass of matrices as a mixture of the atomic distribution at zero and the symmetrized Rayleigh distribution.