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<...
Príomhchruthaitheoirí: | , , , |
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Formáid: | Alt |
Teanga: | English |
Foilsithe / Cruthaithe: |
MDPI AG
2022-05-01
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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. |
first_indexed | 2024-03-10T03:30:21Z |
format | Article |
id | doaj.art-db9ef583f0564d18a187fc2bc2ee1d97 |
institution | Directory Open Access Journal |
issn | 2227-7390 |
language | English |
last_indexed | 2024-03-10T03:30:21Z |
publishDate | 2022-05-01 |
publisher | MDPI AG |
record_format | Article |
series | Mathematics |
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 |
work_keys_str_mv | AT anisoaramariaraducan ontheprobabilityoffindingextremesinarandomset AT constantazoieradulescu ontheprobabilityoffindingextremesinarandomset AT mariusradulescu ontheprobabilityoffindingextremesinarandomset AT gheorghitazbaganu ontheprobabilityoffindingextremesinarandomset |