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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2025-01-01
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| author | Nicola Apollonio |
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| description | 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. |
| format | Article |
| id | doaj-art-2f0c4068bfbd4af584b960c96ce2457d |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
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| series | Axioms |
| spelling | doaj-art-2f0c4068bfbd4af584b960c96ce2457d2025-01-24T13:22:14ZengMDPI AGAxioms2075-16802025-01-011414310.3390/axioms14010043Cantelli’s Bounds for Generalized Tail InequalitiesNicola Apollonio0Istituto per le Applicazioni del Calcolo, C.N.R.,Via dei Taurini 19, 00185 Roma, ItalyLet <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.https://www.mdpi.com/2075-1680/14/1/43tail inequalitiescone dualityWigner matrixnetwork homophily |
| spellingShingle | Nicola Apollonio Cantelli’s Bounds for Generalized Tail Inequalities Axioms tail inequalities cone duality Wigner matrix network homophily |
| title | Cantelli’s Bounds for Generalized Tail Inequalities |
| title_full | Cantelli’s Bounds for Generalized Tail Inequalities |
| title_fullStr | Cantelli’s Bounds for Generalized Tail Inequalities |
| title_full_unstemmed | Cantelli’s Bounds for Generalized Tail Inequalities |
| title_short | Cantelli’s Bounds for Generalized Tail Inequalities |
| title_sort | cantelli s bounds for generalized tail inequalities |
| topic | tail inequalities cone duality Wigner matrix network homophily |
| url | https://www.mdpi.com/2075-1680/14/1/43 |
| work_keys_str_mv | AT nicolaapollonio cantellisboundsforgeneralizedtailinequalities |