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On Tournaments and negative dependence

Part of: Graph theory Multivariate analysis Distribution theory - Probability

Published online by Cambridge University Press:  07 February 2023

Yaakov Malinovsky Open the ORCID record for Yaakov Malinovsky [Opens in a new window]  and
Yosef Rinott
Yaakov Malinovsky*
Affiliation:
University of Maryland, Baltimore County
Yosef Rinott*
Affiliation:
The Hebrew University of Jerusalem
*
*Postal address: Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, USA. Email: yaakovm@umbc.edu
**Postal address: Department of Statistics and Federmann Center for the Study of Rationality, The Hebrew University of Jerusalem, Israel. Email: yosef.rinott@mail.huji.ac.il
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Abstract

Negative dependence of sequences of random variables is often an interesting characteristic of their distribution, as well as a useful tool for studying various asymptotic results, including central limit theorems, Poisson approximations, the rate of increase of the maximum, and more. In the study of probability models of tournaments, negative dependence of participants’ outcomes arises naturally, with application to various asymptotic results. In particular, the property of negative orthant dependence was proved in several articles for different tournament models, with a special proof for each model. In this note we unify these results by proving a stronger property, negative association, a generalization leading to a very simple proof. We also present a natural example of a knockout tournament where the scores are negatively orthant dependent but not negatively associated. The proof requires a new result on a preservation property of negative orthant dependence that is of independent interest.

Keywords

Negative associationnegative orthant dependencemultivariate inequalities

MSC classification

Primary: 62H05: Characterization and structure theory
Secondary: 05C20: Directed graphs (digraphs), tournaments 60E15: Inequalities; stochastic orderings
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 3 , September 2023 , pp. 945 - 954
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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