Details

Autor(en) / Beteiligte

• Götze, F.
• Tikhomirov, A.
• Timushev, D.

Titel

Rate of Convergence for Sparse Sample Covariance Matrices

Ist Teil von

• Foundations of Modern Statistics, p.261-300

Ort / Verlag

Cham: Springer International Publishing

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• This work is devoted to the estimation of the convergence rate of the empirical spectral distribution function (ESD) of sparse sample covariance matrices in the Kolmogorov metric. We consider the case with the sparsity npn∼logαn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$np_n \sim \log ^\alpha n$$\end{document}, for some α>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >1$$\end{document} and assume that the moments of the matrix elements satisfy the condition E|Xjk|4+δ≤C<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{\mathbb {E}}\,}}|X_{jk}|^{4+\delta }\le C<\infty $$\end{document}, for some δ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta >0$$\end{document}. We also obtain approximation estimates for the Stieltjes transform in the bulk.