{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T20:44:41Z","timestamp":1770756281072,"version":"3.50.0"},"reference-count":20,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,6]],"date-time":"2025-01-06T00:00:00Z","timestamp":1736121600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Let X be a centered random vector in a finite-dimensional real inner product space E. For a subset C of the ambient vector space V of E and x,y\u2208V, write x\u2aafCy if y\u2212x\u2208C. If C is a closed convex cone in E, then \u2aafC is a preorder on V, whereas if C is a proper cone in E, then \u2aafC is actually a partial order on V. In this paper, we give sharp Cantelli-type inequalities for generalized tail probabilities such as PrX\u2ab0Cb for b\u2208V. These inequalities are obtained by \u201cscalarizing\u201d X\u2ab0Cb via cone duality and then by minimizing the classical univariate Cantelli\u2019s bound over the scalarized inequalities. Three diverse applications to random matrices, tails of linear images of random vectors, and network homophily are also given.<\/jats:p>","DOI":"10.3390\/axioms14010043","type":"journal-article","created":{"date-parts":[[2025,1,6]],"date-time":"2025-01-06T11:57:58Z","timestamp":1736164678000},"page":"43","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Cantelli\u2019s Bounds for Generalized Tail Inequalities"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6089-1333","authenticated-orcid":false,"given":"Nicola","family":"Apollonio","sequence":"first","affiliation":[{"name":"Istituto per le Applicazioni del Calcolo, C.N.R.,Via dei Taurini 19, 00185 Roma, Italy"},{"name":"Istituto Nazionale di Alta Matematica Francesco Severi, Piazzale Aldo Moro, 5, 00185 Roma, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1214\/aoms\/1177705673","article-title":"Multivariate Chebyshev Inequalities","volume":"31","author":"Marshall","year":"1960","journal-title":"Ann. Math. Stat."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1137\/S0036144504440543","article-title":"Generalized Chebyshev Bounds via Semidefinite Programming","volume":"49","author":"Vandenberghe","year":"2007","journal-title":"SIAM Rev."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Boyd, S., and Vandenberghe, L. (2004). Convex Optimization, Cambridge University Press.","DOI":"10.1017\/CBO9780511804441"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1007\/s004400050119","article-title":"Large deviations for Wigner\u2019s law and Voiculescu\u2019s non-commutative entropy","volume":"108","author":"Arous","year":"1997","journal-title":"Probab. Theory Relat. Fields"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3828","DOI":"10.1093\/imrn\/rnv251","article-title":"What is the probability that a random integral quadratic form in n variables has an integral zero?","volume":"12","author":"Bhargava","year":"2016","journal-title":"Int. Math. Res. Not."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"160201","DOI":"10.1103\/PhysRevLett.97.160201","article-title":"Large deviations of extreme eigenvalues of random matrices","volume":"97","author":"Dean","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1007\/s10687-021-00432-4","article-title":"Tail probabilities of random linear functions of regularly varying random vectors","volume":"25","author":"Das","year":"2022","journal-title":"Extremes"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"9594","DOI":"10.1093\/imrn\/rnu243","article-title":"Small Ball Probabilities for Linear Images of High-Dimensional Distributions","volume":"19","author":"Rudelson","year":"2015","journal-title":"Int. Math. Res. Not."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"054130","DOI":"10.1103\/PhysRevE.108.054130","article-title":"Network homophily via tail inequalities","volume":"108","author":"Apollonio","year":"2023","journal-title":"Phys. Rev. E"},{"key":"ref_10","unstructured":"Shaked-Monderer, N., and Berman, A. (2021). Copositive and Completely Positive Matrices, World Scientific Publishing."},{"key":"ref_11","unstructured":"Schneider, R. (2013). Convex Bodies: The Brunn\u2013Minkowski Theory, Cambridge University Press. Encyclopedia of Mathematics and its Applications."},{"key":"ref_12","unstructured":"Schrijver, A. (1986). Theory of Linear and Integer Programming, Wiley."},{"key":"ref_13","unstructured":"Horn, R., and Johnson, C. (2013). Matrix Analysis, Cambridge University Press. [2nd ed.]."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Berman, A., and Plemmons, R.J. (1994). Nonnegative Matrices in the Mathematical Sciences, SIAM Press.","DOI":"10.1137\/1.9781611971262"},{"key":"ref_15","unstructured":"Sanz-Sol, M., Soria, J., Varona, J.L., and Verdera, J. (2006, January 22\u201330). Advances in convex optimization: Conic programming. Proceedings of the International Congress of Mathematicians, Madrid, Spain."},{"key":"ref_16","unstructured":"Eaton, M.L. (1983). Multivariate Statistics: A Vector Space Approach, Wiley."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1017\/S0962492904000236","article-title":"Random matrix theory","volume":"14","author":"Edelman","year":"2005","journal-title":"Acta Numer."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"298","DOI":"10.21136\/CMJ.1973.101168","article-title":"Algebraic connectivity of graphs","volume":"23","author":"Fiedler","year":"1973","journal-title":"Czechoslov. Math. J."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.dam.2024.04.025","article-title":"Second-order moments of the size of randomly induced subgraphs of given order","volume":"355","author":"Apollonio","year":"2024","journal-title":"Discret. Appl. Math."},{"key":"ref_20","unstructured":"Garey, M.R., and Johnson, D.S. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman and Company."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/43\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:23:54Z","timestamp":1759919034000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/43"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,6]]},"references-count":20,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["axioms14010043"],"URL":"https:\/\/doi.org\/10.3390\/axioms14010043","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,6]]}}}