The circular law asserts that the spectral measure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point z away from the unit circle. More precisely, if ||z| − 1| > τ for arbitrarily small τ > 0, the circular law is valid around z up to scale N−1/2+ε for any ε > 0 under the assu...

In this paper, we study the asymptotic theory for spectral analysis of stationary random fields on Z2, including linear and nonlinear fields. Asymptotic properties Fourier coefficients periodograms, limiting distributions coefficients, uniform consistency kernel density estimators are obtained under various mild conditions moments dependence structures. The validity aforementioned results estim...

Let $W_n := \sum_{j=1}^n Z_j$ be a sum of independent integer-valued random variables. In this paper, we derive an asymptotic expansion for the probability $\mathbb{P}[W_n \in A]$ of an arbitrary subset $A \in \mathbb{Z}$. Our approximation improves upon the classical expansions by including an explicit, uniform error estimate, involving only easily computable properties of the distributions of...