<p>For a sequence <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1709_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">$\{X_{n}, \, n \geqslant 1 \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>,</mo> <mspace width="0.166667em" /> <mi>n</mi> <mo>⩾</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> of random variables satisfying <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1709_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {E}} |X_{n} |< \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="double-struck">E</mi> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">|</mo> <mo><</mo> <mi>∞</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1709_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n \geqslant 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, a maximal inequality is established, and used to obtain strong law of large numbers for dependent random variables.</p>

A maximal inequality for dependent random variables

期刊论文
发表日期：

2025年2月20日

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João Lita da Silva

摘要

For a sequence $\{X_{n}, \, n \geqslant 1 \}$ ${X_{n}, n ⩾ 1}$ of random variables satisfying ${\mathbb {E}} |X_{n} |< \infty $ $E | X_{n} | < \infty$ for all $n \geqslant 1$ $n ⩾ 1$ , a maximal inequality is established, and used to obtain strong law of large numbers for dependent random variables.