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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
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| Series: | Entropy |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1099-4300/27/1/41 |
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| Summary: | 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. |
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| ISSN: | 1099-4300 |