Local Marchenko–Pastur Law
for Sparse Rectangular Random Matrices
F. Götzea,*, D. A. Timushevb,**, and A. N. Tikhomirov b,***
a Bielefeld University, Bielefeld, Germany
b Komi Research Center, Ural Branch,
Russian Academy of Sciences, Syktyvkar, Russia
Correspondence to: *e-mail: goetze@math.uni-bielefeld.de
Correspondence to: **e-mail: timushev@ipm.komisc.ru
Correspondence to: ***e-mail: tikhomirov@ipm.komisc.ru
Received 9 September, 2021
Abstract—We consider sparse sample covariance matrices with sparsity probability ${{p}_{n}} \geqslant {{c}_{0}}{{\log }^{{\frac{2}{\varkappa }}}}n{\text{/}}n$ with $\varkappa > 0$. Assuming that the distribution of matrix elements has a finite absolute moment of order $4 + \delta $, $\delta > 0$, it is shown that the distance between the Stieltjes transforms of the empirical spectral distribution function and the Marchenko–Pastur law is of order $\log n(1{\text{/}}(nv) + 1{\text{/}}(n{{p}_{n}}))$, where v is the distance to the real axis in the complex plane.
Keywords: local Marchenko–Pastur law, local regime, sparse random matrices, spectrum of a random matrix, Stieltjes transform
DOI: 10.1134/S1064562421060065