Cantelli’s Bounds for Generalized Tail Inequalities
Let <i>X</i> be a centered random vector in a finite-dimensional real inner product space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">E</mi></semantics...
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/1/43 |
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| Summary: | Let <i>X</i> be a centered random vector in a finite-dimensional real inner product space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">E</mi></semantics></math></inline-formula>. For a subset <i>C</i> of the ambient vector space <i>V</i> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">E</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>y</mi><mo>∈</mo><mi>V</mi></mrow></semantics></math></inline-formula>, write <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><msub><mo>⪯</mo><mi>C</mi></msub><mi>y</mi></mrow></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>−</mo><mi>x</mi><mo>∈</mo><mi>C</mi></mrow></semantics></math></inline-formula>. If <i>C</i> is a closed convex cone in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">E</mi></semantics></math></inline-formula>, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>⪯</mo><mi>C</mi></msub></semantics></math></inline-formula> is a preorder on <i>V</i>, whereas if <i>C</i> is a proper cone in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">E</mi></semantics></math></inline-formula>, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>⪯</mo><mi>C</mi></msub></semantics></math></inline-formula> is actually a partial order on <i>V</i>. In this paper, we give sharp Cantelli-type inequalities for generalized tail probabilities such as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Pr</mi><mfenced separators="" open="{" close="}"><mi>X</mi><msub><mo>⪰</mo><mi>C</mi></msub><mi>b</mi></mfenced></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>V</mi></mrow></semantics></math></inline-formula>. These inequalities are obtained by “scalarizing” <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><msub><mo>⪰</mo><mi>C</mi></msub><mi>b</mi></mrow></semantics></math></inline-formula> via cone duality and then by minimizing the classical univariate Cantelli’s bound over the scalarized inequalities. Three diverse applications to random matrices, tails of linear images of random vectors, and network homophily are also given. |
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| ISSN: | 2075-1680 |