On the Probability of Finding Extremes in a Random Set

We consider a sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mfenced separators="" open="(" close=")"><msub><mi mathvariant="bold">Z<...

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Príomhchruthaitheoirí: Anișoara Maria Răducan, Constanța Zoie Rădulescu, Marius Rădulescu, Gheorghiță Zbăganu
Formáid: Alt
Teanga:English
Foilsithe / Cruthaithe: MDPI AG 2022-05-01
Sraith:Mathematics
Ábhair:
Rochtain ar líne:https://www.mdpi.com/2227-7390/10/10/1623
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author Anișoara Maria Răducan
Constanța Zoie Rădulescu
Marius Rădulescu
Gheorghiță Zbăganu
author_facet Anișoara Maria Răducan
Constanța Zoie Rădulescu
Marius Rădulescu
Gheorghiță Zbăganu
author_sort Anișoara Maria Răducan
collection DOAJ
description We consider a sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mfenced separators="" open="(" close=")"><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub></mfenced><mrow><mi>j</mi><mo>≥</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> of i.i.d. <i>d</i>-dimensional random vectors and for every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> consider the sample <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">S</mi><mi>n</mi></msub><mo>=</mo><mrow><mo>{</mo><msub><mi mathvariant="bold">Z</mi><mn>1</mn></msub><mo>,</mo><msub><mi mathvariant="bold">Z</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi mathvariant="bold">Z</mi><mi>n</mi></msub><mo>}</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> We say that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub></semantics></math></inline-formula> is a “leader” in the sample <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub><mo>≥</mo><msub><mi mathvariant="bold">Z</mi><mi>k</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi>k</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo><mspace width="0.166667em"></mspace><mspace width="4pt"></mspace></mrow></semantics></math></inline-formula>and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub></semantics></math></inline-formula> is an “anti-leader” if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub><mo>≤</mo><msub><mi mathvariant="bold">Z</mi><mi>k</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi>k</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></mrow></semantics></math></inline-formula>. After all, the leader and the anti-leader are the naive extremes. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mi>n</mi></msub></semantics></math></inline-formula> be the probability that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> has a leader, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>b</mi><mi>n</mi></msub></semantics></math></inline-formula> be the probability that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> has an anti-leader and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mi>n</mi></msub></semantics></math></inline-formula> be the probability that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> has both a leader and an anti-leader. One of the aims of the paper is to compute, or, at least to estimate, or if even that is not possible, to estimate the limits of this quantities. Another goal is to find conditions on the distribution <i>F</i> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mfenced separators="" open="(" close=")"><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub></mfenced><mrow><mi>j</mi><mo>≥</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> so that the inferior limits of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>,</mo><msub><mi>b</mi><mi>n</mi></msub><mo>,</mo><msub><mi>c</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> are positive. We give examples of distributions for which we can compute these probabilities and also examples when we are not able to do that. Then we establish conditions, unfortunately only sufficient when the limits are positive. Doing that we discovered a lot of open questions and we make two annoying conjectures—annoying because they seemed to be obvious but at a second thought we were not able to prove them. It seems that these problems have never been approached in the literature.
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spelling doaj.art-db9ef583f0564d18a187fc2bc2ee1d972023-11-23T11:59:59ZengMDPI AGMathematics2227-73902022-05-011010162310.3390/math10101623On the Probability of Finding Extremes in a Random SetAnișoara Maria Răducan0Constanța Zoie Rădulescu1Marius Rădulescu2Gheorghiță Zbăganu3“Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, RomaniaNational Institute for Research and Development in Informatics, 011455 Bucharest, Romania“Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, Romania“Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, RomaniaWe consider a sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mfenced separators="" open="(" close=")"><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub></mfenced><mrow><mi>j</mi><mo>≥</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> of i.i.d. <i>d</i>-dimensional random vectors and for every <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> consider the sample <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">S</mi><mi>n</mi></msub><mo>=</mo><mrow><mo>{</mo><msub><mi mathvariant="bold">Z</mi><mn>1</mn></msub><mo>,</mo><msub><mi mathvariant="bold">Z</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi mathvariant="bold">Z</mi><mi>n</mi></msub><mo>}</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> We say that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub></semantics></math></inline-formula> is a “leader” in the sample <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub><mo>≥</mo><msub><mi mathvariant="bold">Z</mi><mi>k</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi>k</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo><mspace width="0.166667em"></mspace><mspace width="4pt"></mspace></mrow></semantics></math></inline-formula>and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub></semantics></math></inline-formula> is an “anti-leader” if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub><mo>≤</mo><msub><mi mathvariant="bold">Z</mi><mi>k</mi></msub><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi>k</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></mrow></semantics></math></inline-formula>. After all, the leader and the anti-leader are the naive extremes. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>a</mi><mi>n</mi></msub></semantics></math></inline-formula> be the probability that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> has a leader, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>b</mi><mi>n</mi></msub></semantics></math></inline-formula> be the probability that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> has an anti-leader and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mi>n</mi></msub></semantics></math></inline-formula> be the probability that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">S</mi><mi>n</mi></msub></semantics></math></inline-formula> has both a leader and an anti-leader. One of the aims of the paper is to compute, or, at least to estimate, or if even that is not possible, to estimate the limits of this quantities. Another goal is to find conditions on the distribution <i>F</i> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mfenced separators="" open="(" close=")"><msub><mi mathvariant="bold">Z</mi><mi>j</mi></msub></mfenced><mrow><mi>j</mi><mo>≥</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> so that the inferior limits of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>,</mo><msub><mi>b</mi><mi>n</mi></msub><mo>,</mo><msub><mi>c</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> are positive. We give examples of distributions for which we can compute these probabilities and also examples when we are not able to do that. Then we establish conditions, unfortunately only sufficient when the limits are positive. Doing that we discovered a lot of open questions and we make two annoying conjectures—annoying because they seemed to be obvious but at a second thought we were not able to prove them. It seems that these problems have never been approached in the literature.https://www.mdpi.com/2227-7390/10/10/1623stochastic orderrandom vectormultivariate distributions
spellingShingle Anișoara Maria Răducan
Constanța Zoie Rădulescu
Marius Rădulescu
Gheorghiță Zbăganu
On the Probability of Finding Extremes in a Random Set
Mathematics
stochastic order
random vector
multivariate distributions
title On the Probability of Finding Extremes in a Random Set
title_full On the Probability of Finding Extremes in a Random Set
title_fullStr On the Probability of Finding Extremes in a Random Set
title_full_unstemmed On the Probability of Finding Extremes in a Random Set
title_short On the Probability of Finding Extremes in a Random Set
title_sort on the probability of finding extremes in a random set
topic stochastic order
random vector
multivariate distributions
url https://www.mdpi.com/2227-7390/10/10/1623
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