This article is cited in 1 paper

MATHEMATICS

Local Marchenko–Pastur law for sparse rectangular random matrices

F. Götze^a, D. A. Timushev^b, A. N. Tikhomirov<sup>b

a</sup> Bielefeld University, Bielefeld, Germany
^b Komi Scientific Center of Ural Branch of RAS, Syktyvkar, Russia

Abstract: We consider sparse sample covariance matrices with sparsity probability $p_n\ge c_0\log^{\frac2\kappa}n/n$ with $\kappa>0$. 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 $\log n(1/(nv)+1/(np_n))$, где 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.

UDC: 519.2

Presented: I. A. Ibragimov
Received: 09.09.2021
Revised: 09.09.2021
Accepted: 27.10.2021

DOI: 10.31857/S2686954321060060

 English version: Doklady Mathematics, 2021, 104:3, 332–335

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