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<p>In this study, we explore both weak and strong convergence properties of extreme eigenvalues in a large-dimensional sample covariance matrix, specifically in cases where the data matrix comprises independent, though not identically distributed, elements. Our findings reveal that, provided there exists a uniform boundedness condition on the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1415_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\((2+\delta )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-th moment for some <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1415_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\delta &gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and the proper Lindeberg condition is satisfied, the established convergence results in Yin, Bai and Krishnaiah, (Probab. Theory Relat. Fields 78:509–521, 1988) and Bai, and Yin (Ann. Probab. 21:1275–1294, 1993) remain applicable.</p>

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A Note on the Convergence of the Extreme Eigenvalues of a Large-Dimensional Sample Covariance Matrix

  • 期刊论文

  • 发表日期：

    2025年4月22日

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  • Shizhe Hong,
  • Haiyan Song,
  • Jiang Hu,
  • Zhidong Bai

摘要

In this study, we explore both weak and strong convergence properties of extreme eigenvalues in a large-dimensional sample covariance matrix, specifically in cases where the data matrix comprises independent, though not identically distributed, elements. Our findings reveal that, provided there exists a uniform boundedness condition on the \((2+\delta )\) ( 2 + δ ) -th moment for some \(\delta >0\) δ > 0 and the proper Lindeberg condition is satisfied, the established convergence results in Yin, Bai and Krishnaiah, (Probab. Theory Relat. Fields 78:509–521, 1988) and Bai, and Yin (Ann. Probab. 21:1275–1294, 1993) remain applicable.