CLT for generalized patterned random matrices: a unified approach

In this paper, we derive a unified method for establishing the distributional convergence of linear eigenvalue statistics (LES) for generalized patterned random matrices. We prove that for an $N \times N$ generalized patterned random matrix with independent subexponential entries and even degree monomial test functions of degree $p_n=o(\log N/\log \log N)$, the LES converges to standard Gaussian distribution. This generalizes the CLT results on Gaussian patterned random matrices in Chatterjee(2009), Adhikari and Saha(2017). As an application, new results on LES of Toeplitz, Hankel, circulant-type matrices and block patterned random matrices for varying test functions are derived. For odd degree monomial test functions, we derive the limiting moments of LES and show that it may not converge to a Gaussian distribution.