Project Euclid - mathematics and statistics online

We proved the local Marchenko–Pastur law for sparse sample covariance matrices that corresponded to rectangular observation matrices of order $n\times m$ with $n/m\to y$ (where $y>0$) and sparse probability $n{p}_n>{log}^{\beta }n$ (where $\beta >0$). The bounds of the distance between the empirical spectral distribution function of the sparse sample covariance matrices and the Marchenko–Pastur law distribution function that was obtained in the complex domain $z\in \mathcal{D}$ with $Imz>v_0>0$ (where $v_0$) were of order ${log}^{4}n/n$ and the domain bounds did not depend on $p_n$ while $n{p}_n>{log}^{\beta }n$.

We present the results from laser granulometry of a nanosized fraction of aerosol substance and from UV spectrometry of water washouts from different-age needles in four plant species. It is estimated how the hydrothermal conditions influence the time variations in morphometric indices of needles and contamination of their surfaces by phenolic compounds in the composition of secondary organic aerosols. We discuss how secondary organic aerosols are generated on the surfaces of different-age needles during efflorescence of phenolic compounds and their subsequent entry into the forest canopy during turbulent separation of the boundary layer from the leaf surface.

This paper investigates the spectral properties of block circulant matrices with high-order symmetric (or Hermitian) blocks. We analyze cases with dependent or sparse independent entries within these blocks. Additionally, we analyze the distribution of singular values for the product of independent circulant matrices with non-Hermitian blocks.