Self-Normalized Moderate Deviations for Degenerate <i>U</i>-Statistics
In this paper, we study self-normalized moderate deviations for degenerate <i>U</i>-statistics of order 2. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub>...
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2025-01-01
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| author | Lin Ge Hailin Sang Qi-Man Shao |
| author_facet | Lin Ge Hailin Sang Qi-Man Shao |
| author_sort | Lin Ge |
| collection | DOAJ |
| description | In this paper, we study self-normalized moderate deviations for degenerate <i>U</i>-statistics of order 2. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>X</mi><mi>i</mi></msub><mo>,</mo><mi>i</mi><mo>≥</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula> be i.i.d. random variables and consider symmetric and degenerate kernel functions in the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup><msub><mi>λ</mi><mi>l</mi></msub><msub><mi>g</mi><mi>l</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msub><mi>g</mi><mi>l</mi></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mi>l</mi></msub><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><msub><mi>g</mi><mi>l</mi></msub><mrow><mo>(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>g</mi><mi>l</mi></msub><mrow><mo>(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is in the domain of attraction of a normal law for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Under the condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup><msub><mi>λ</mi><mi>l</mi></msub><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula> and some truncated conditions for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>g</mi><mi>l</mi></msub><mrow><mo>(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>:</mo><mi>l</mi><mo>≥</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula>, we show that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>log</mi><mi>P</mi><mrow><mo>(</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><msub><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≠</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub><mi>h</mi><mrow><mo>(</mo><msub><mi>X</mi><mi>i</mi></msub><mo>,</mo><msub><mi>X</mi><mi>j</mi></msub><mo>)</mo></mrow></mrow><mrow><msub><mi>max</mi><mrow><mn>1</mn><mo>≤</mo><mi>l</mi><mo><</mo><mo>∞</mo></mrow></msub><msub><mi>λ</mi><mi>l</mi></msub><msubsup><mi>V</mi><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow><mn>2</mn></msubsup></mrow></mfrac></mstyle><mo>≥</mo><msubsup><mi>x</mi><mi>n</mi><mn>2</mn></msubsup><mo>)</mo></mrow><mo>∼</mo><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msubsup><mi>x</mi><mi>n</mi><mn>2</mn></msubsup><mn>2</mn></mfrac></mstyle></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>=</mo><mi>o</mi><mrow><mo>(</mo><msqrt><mi>n</mi></msqrt><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>V</mi><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow><mn>2</mn></msubsup><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msubsup><mi>g</mi><mi>l</mi><mn>2</mn></msubsup><mrow><mo>(</mo><msub><mi>X</mi><mi>i</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. As application, a law of the iterated logarithm is also obtained. |
| format | Article |
| id | doaj-art-39eebc8432cb4b389afc70f96a66cdd6 |
| institution | Kabale University |
| issn | 1099-4300 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj-art-39eebc8432cb4b389afc70f96a66cdd62025-01-24T13:31:46ZengMDPI AGEntropy1099-43002025-01-012714110.3390/e27010041Self-Normalized Moderate Deviations for Degenerate <i>U</i>-StatisticsLin Ge0Hailin Sang1Qi-Man Shao2Division of Arts and Sciences, Mississippi State University at Meridian, Meridian, MS 39307, USADepartment of Mathematics, University of Mississippi, University, MS 38677, USADepartment of Statistics and Data Science, Shenzhen International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, ChinaIn this paper, we study self-normalized moderate deviations for degenerate <i>U</i>-statistics of order 2. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>X</mi><mi>i</mi></msub><mo>,</mo><mi>i</mi><mo>≥</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula> be i.i.d. random variables and consider symmetric and degenerate kernel functions in the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup><msub><mi>λ</mi><mi>l</mi></msub><msub><mi>g</mi><mi>l</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msub><mi>g</mi><mi>l</mi></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mi>l</mi></msub><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><msub><mi>g</mi><mi>l</mi></msub><mrow><mo>(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>g</mi><mi>l</mi></msub><mrow><mo>(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is in the domain of attraction of a normal law for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Under the condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup><msub><mi>λ</mi><mi>l</mi></msub><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula> and some truncated conditions for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>g</mi><mi>l</mi></msub><mrow><mo>(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>:</mo><mi>l</mi><mo>≥</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula>, we show that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>log</mi><mi>P</mi><mrow><mo>(</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><msub><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≠</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub><mi>h</mi><mrow><mo>(</mo><msub><mi>X</mi><mi>i</mi></msub><mo>,</mo><msub><mi>X</mi><mi>j</mi></msub><mo>)</mo></mrow></mrow><mrow><msub><mi>max</mi><mrow><mn>1</mn><mo>≤</mo><mi>l</mi><mo><</mo><mo>∞</mo></mrow></msub><msub><mi>λ</mi><mi>l</mi></msub><msubsup><mi>V</mi><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow><mn>2</mn></msubsup></mrow></mfrac></mstyle><mo>≥</mo><msubsup><mi>x</mi><mi>n</mi><mn>2</mn></msubsup><mo>)</mo></mrow><mo>∼</mo><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msubsup><mi>x</mi><mi>n</mi><mn>2</mn></msubsup><mn>2</mn></mfrac></mstyle></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>=</mo><mi>o</mi><mrow><mo>(</mo><msqrt><mi>n</mi></msqrt><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>V</mi><mrow><mi>n</mi><mo>,</mo><mi>l</mi></mrow><mn>2</mn></msubsup><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msubsup><mi>g</mi><mi>l</mi><mn>2</mn></msubsup><mrow><mo>(</mo><msub><mi>X</mi><mi>i</mi></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. As application, a law of the iterated logarithm is also obtained.https://www.mdpi.com/1099-4300/27/1/41moderate deviationdegenerate <i>U</i>-statisticslaw of the iterated logarithmself-normalization |
| spellingShingle | Lin Ge Hailin Sang Qi-Man Shao Self-Normalized Moderate Deviations for Degenerate <i>U</i>-Statistics Entropy moderate deviation degenerate <i>U</i>-statistics law of the iterated logarithm self-normalization |
| title | Self-Normalized Moderate Deviations for Degenerate <i>U</i>-Statistics |
| title_full | Self-Normalized Moderate Deviations for Degenerate <i>U</i>-Statistics |
| title_fullStr | Self-Normalized Moderate Deviations for Degenerate <i>U</i>-Statistics |
| title_full_unstemmed | Self-Normalized Moderate Deviations for Degenerate <i>U</i>-Statistics |
| title_short | Self-Normalized Moderate Deviations for Degenerate <i>U</i>-Statistics |
| title_sort | self normalized moderate deviations for degenerate i u i statistics |
| topic | moderate deviation degenerate <i>U</i>-statistics law of the iterated logarithm self-normalization |
| url | https://www.mdpi.com/1099-4300/27/1/41 |
| work_keys_str_mv | AT linge selfnormalizedmoderatedeviationsfordegenerateiuistatistics AT hailinsang selfnormalizedmoderatedeviationsfordegenerateiuistatistics AT qimanshao selfnormalizedmoderatedeviationsfordegenerateiuistatistics |