<p>We establish the Strassen’s law of the iterated logarithm (LIL for short) for independent and identically distributed random variables with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_2759_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="132" /> </InlineMediaObject> <EquationSource Format="TEX">${\hat {\mathbb E}}[X_{1}]={\hat {\cal E}}[X_{1}]=0$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mover> <mrow> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mrow> <mrow> <mover> <mi mathvariant="script">E</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_2759_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$C_{\mathbb V}[X_{1}^{2}]<\infty$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>C</mi> <mrow> <mrow> <mi mathvariant="double-struck">V</mi> </mrow> </mrow> </msub> <mo stretchy="false">[</mo> <msubsup> <mi>X</mi> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> </math></EquationSource> </InlineEquation> under a sub-linear expectation space with a countably sub-additive capacity <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_2759_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb V}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="double-struck">V</mi> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We also show the LIL for upper capacity with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_2759_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma={\overline \sigma}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>σ</mi> <mo>=</mo> <mrow> <mover> <mi>σ</mi> <mo accent="false">¯</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> under some certain conditions.</p>

Strassen’s Law of the Iterated Logarithm under Sub-linear Expectations

期刊论文
发表日期：

2025年1月24日

查看全文

Wangyun Gu,
Lixin Zhang

摘要

We establish the Strassen’s law of the iterated logarithm (LIL for short) for independent and identically distributed random variables with ${\hat {\mathbb E}}[X_{1}]={\hat {\cal E}}[X_{1}]=0$ $\hat{E} [X_{1}] = \hat{E} [X_{1}] = 0$ and $C_{\mathbb V}[X_{1}^{2}]<\infty$ $C_{V} [X_{1}^{2}] < \infty$ under a sub-linear expectation space with a countably sub-additive capacity ${\mathbb V}$ $V$ . We also show the LIL for upper capacity with $\sigma={\overline \sigma}$ $σ = \bar{σ}$ under some certain conditions.