Near-Record Values in Discrete Random Sequences
Given a sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/14/2442 |
| _version_ | 1827605145627656192 |
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| author | Miguel Lafuente Raúl Gouet F. Javier López Gerardo Sanz |
| author_facet | Miguel Lafuente Raúl Gouet F. Javier López Gerardo Sanz |
| author_sort | Miguel Lafuente |
| collection | DOAJ |
| description | Given a sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> of random variables, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>X</mi><mi>n</mi></msub></semantics></math></inline-formula> is said to be a near-record if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>n</mi></msub><mo>∈</mo><mrow><mo>(</mo><msub><mi>M</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><mi>a</mi><mo>,</mo><msub><mi>M</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>]</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mi>n</mi></msub><mo>=</mo><mo movablelimits="true" form="prefix">max</mo><mrow><mo>{</mo><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>X</mi><mi>n</mi></msub><mo>}</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> is a parameter. We investigate the point process <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></semantics></math></inline-formula> of near-record values from an integer-valued, independent and identically distributed sequence, showing that it is a Bernoulli cluster process. We derive the probability generating functional of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula> and formulas for the expectation, variance and covariance of the counting variables <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>,</mo><mi>A</mi><mo>⊂</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We also derive the strong convergence and asymptotic normality of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>]</mo><mo>)</mo></mrow></semantics></math></inline-formula>, as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></semantics></math></inline-formula>, under mild regularity conditions on the distribution of the observations. For heavy-tailed distributions, with square-summable hazard rates, we prove that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>]</mo><mo>)</mo></mrow></semantics></math></inline-formula> grows to a finite random limit and compute its probability generating function. We present examples of the application of our results to particular distributions, covering a wide range of behaviours in terms of their right tails. |
| first_indexed | 2024-03-09T06:15:34Z |
| format | Article |
| id | doaj.art-5d107c053ea54865898003a25d029cd7 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T06:15:34Z |
| publishDate | 2022-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-5d107c053ea54865898003a25d029cd72023-12-03T11:53:40ZengMDPI AGMathematics2227-73902022-07-011014244210.3390/math10142442Near-Record Values in Discrete Random SequencesMiguel Lafuente0Raúl Gouet1F. Javier López2Gerardo Sanz3Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, C/Pedro Cerbuna 12, 50009 Zaragoza, SpainDepartamento Ingeniería Matemática y Centro de Modelamiento Matemático (CNRS IRL 2807), Universidad de Chile, Avenida Beauchef 851, Santiago 8370456, ChileDepartamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, C/Pedro Cerbuna 12, 50009 Zaragoza, SpainDepartamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, C/Pedro Cerbuna 12, 50009 Zaragoza, SpainGiven a sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> of random variables, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>X</mi><mi>n</mi></msub></semantics></math></inline-formula> is said to be a near-record if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mi>n</mi></msub><mo>∈</mo><mrow><mo>(</mo><msub><mi>M</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><mi>a</mi><mo>,</mo><msub><mi>M</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>]</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mi>n</mi></msub><mo>=</mo><mo movablelimits="true" form="prefix">max</mo><mrow><mo>{</mo><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>X</mi><mi>n</mi></msub><mo>}</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> is a parameter. We investigate the point process <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></semantics></math></inline-formula> of near-record values from an integer-valued, independent and identically distributed sequence, showing that it is a Bernoulli cluster process. We derive the probability generating functional of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula> and formulas for the expectation, variance and covariance of the counting variables <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>,</mo><mi>A</mi><mo>⊂</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We also derive the strong convergence and asymptotic normality of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>]</mo><mo>)</mo></mrow></semantics></math></inline-formula>, as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></semantics></math></inline-formula>, under mild regularity conditions on the distribution of the observations. For heavy-tailed distributions, with square-summable hazard rates, we prove that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>]</mo><mo>)</mo></mrow></semantics></math></inline-formula> grows to a finite random limit and compute its probability generating function. We present examples of the application of our results to particular distributions, covering a wide range of behaviours in terms of their right tails.https://www.mdpi.com/2227-7390/10/14/2442recordnear-recordBernoulli cluster processlaw of large numberscentral limit theorem |
| spellingShingle | Miguel Lafuente Raúl Gouet F. Javier López Gerardo Sanz Near-Record Values in Discrete Random Sequences Mathematics record near-record Bernoulli cluster process law of large numbers central limit theorem |
| title | Near-Record Values in Discrete Random Sequences |
| title_full | Near-Record Values in Discrete Random Sequences |
| title_fullStr | Near-Record Values in Discrete Random Sequences |
| title_full_unstemmed | Near-Record Values in Discrete Random Sequences |
| title_short | Near-Record Values in Discrete Random Sequences |
| title_sort | near record values in discrete random sequences |
| topic | record near-record Bernoulli cluster process law of large numbers central limit theorem |
| url | https://www.mdpi.com/2227-7390/10/14/2442 |
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