Let {X(t)}t⩾0 be the generalized fractional Brownian motion introduced by Pang and Taqqu (2019): <Equation ID="Equ1"> <EquationSource Format="TEX">${\{ X(t)\} _{t \geqslant 0}}\mathop = \limits^{\text{d}} {\left\{ {\int_\mathbb{R} {\left( {(t - u)_ + ^\alpha - ( - u)_ + ^\alpha } \right)} |u{|^{ - \gamma /2}}B({\text{d}}u)} \right\}_{t \geqslant 0}},$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mtable> <mtr> <mtd> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <mover> <mo>=</mo> <mi mathvariant="normal">d</mi> </mover> </mtd> <mtd> <msub> <mrow> <mo>{</mo> <msub> <mo>∫</mo> <mrow> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>u</mi> <msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mi>α</mi> </mrow> </msubsup> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>u</mi> <msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mi>α</mi> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>u</mi> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>−</mo> <mi>γ</mi> <mrow> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mi>B</mi> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="normal">d</mi> </mrow> <mi>u</mi> <mo stretchy="false">)</mo> <mo>}</mo> </mrow> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mtd> </mtr> </mtable> </math></EquationSource> </Equation> where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\gamma \in [0,1),\;\;\alpha \in \left( { - {1 \over 2} + {\gamma \over 2},\;{1 \over 2} + {\gamma \over 2}} \right)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>γ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>α</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow> <mfrac> <mi>γ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </math></EquationSource> </InlineEquation> are constants. For any θ > 0, let <Equation ID="Equ2"> <EquationSource Format="TEX">$Y(t) = \frac{1}{{\Gamma (\theta )}}\int_0^t {{{(t - u)}^{\theta - 1}}} X(u){\text{d}}u,\quad t \geqslant 0.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mtable> <mtr> <mtd> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>t</mi> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>u</mi> <msup> <mo stretchy="false">)</mo> <mrow> <mi>θ</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>X</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mrow> <mi mathvariant="normal">d</mi> </mrow> <mi>u</mi> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>≥</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </math></EquationSource> </Equation>Building upon the argument of Talagrand (1996), we develop integral criteria characterizing the lower classes of the process Y at t = 0 and at infinity. As a consequence, we derive its Chung-type laws of the iterated logarithm. In the proofs, the small ball probability estimates play important roles.

Lower classes and Chung’s LILs of the fractional integrated generalized fractional Brownian motion

期刊论文
发表日期：

2026年5月9日

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Mengjie Lyu,
Min Wang,
Ran Wang

摘要

Let {X(t)}_t⩾0 be the generalized fractional Brownian motion introduced by Pang and Taqqu (2019): ${\{ X(t)\} _{t \geqslant 0}}\mathop = \limits^{\text{d}} {\left\{ {\int_\mathbb{R} {\left( {(t - u)_ + ^\alpha - ( - u)_ + ^\alpha } \right)} |u{|^{ - \gamma /2}}B({\text{d}}u)} \right\}_{t \geqslant 0}},$ $\begin{matrix} {X (t)}_{t \geq 0} \overset{d}{=} & {\int_{R} ((t - u)_{+}^{α} - (- u)_{+}^{α}) | u |^{- γ / 2} B (d u)}_{t \geq 0}, \end{matrix}$ where $\gamma \in [0,1),\;\;\alpha \in \left( { - {1 \over 2} + {\gamma \over 2},\;{1 \over 2} + {\gamma \over 2}} \right)$ $γ \in [0, 1), α \in (- \frac{1}{2} + \frac{γ}{2}, \frac{1}{2} + \frac{γ}{2})$ are constants. For any θ > 0, let $Y(t) = \frac{1}{{\Gamma (\theta )}}\int_0^t {{{(t - u)}^{\theta - 1}}} X(u){\text{d}}u,\quad t \geqslant 0.$ $\begin{matrix} Y (t) = \frac{1}{Γ (θ)} \int_{0}^{t} (t - u)^{θ - 1} X (u) d u, t \geq 0. \end{matrix}$

Building upon the argument of Talagrand (1996), we develop integral criteria characterizing the lower classes of the process Y at t = 0 and at infinity. As a consequence, we derive its Chung-type laws of the iterated logarithm. In the proofs, the small ball probability estimates play important roles.