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{"title":"更新过程的某些首次通过时间和区域的渐近结果","authors":"C. Macci, B. Pacchiarotti","doi":"10.1090/tpms/1189","DOIUrl":null,"url":null,"abstract":"<p>We consider the process <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartSet x minus upper N left-parenthesis t right-parenthesis colon t greater-than-or-equal-to 0 EndSet\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\{x-N(t):t\\geq 0\\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"x element-of double-struck upper R Subscript plus\"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">x\\in \\mathbb {R}_+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartSet upper N left-parenthesis t right-parenthesis colon t greater-than-or-equal-to 0 EndSet\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\{N(t):t\\geq 0\\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis tau left-parenthesis x right-parenthesis comma upper A left-parenthesis x right-parenthesis right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(\\tau (x),A(x))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"tau left-parenthesis x right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\tau (x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the first-passage time of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartSet x minus upper N left-parenthesis t right-parenthesis colon t greater-than-or-equal-to 0 EndSet\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\{x-N(t):t\\geq 0\\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to reach zero or a negative value, and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A left-parenthesis x right-parenthesis colon-equal integral Subscript 0 Superscript tau left-parenthesis x right-parenthesis Baseline left-parenthesis x minus upper N left-parenthesis t right-parenthesis right-parenthesis d t\"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>≔</mml:mo> <mml:msubsup> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mn>0</mml:mn> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">A(x)≔\\int _0^{\\tau (x)}(x-N(t))dt</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the corresponding first-passage (positive) area swept out by the process <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartSet x minus upper N left-parenthesis t right-parenthesis colon t greater-than-or-equal-to 0 EndSet\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\{x-N(t):t\\geq 0\\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We remark that we can define the sequence <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartSet left-parenthesis tau left-parenthesis n right-parenthesis comma upper A left-parenthesis n right-parenthesis right-parenthesis colon n greater-than-or-equal-to 1 EndSet\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\{(\\tau (n),A(n)):n\\geq 1\\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by referring to the concept of integrated random walk. Our aim is to prove asymptotic results as <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"x right-arrow normal infinity\"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">→<!-- → --></mml:mo> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">x\\to \\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the fashion of large (and moderate) deviations.</p>","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2021-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Asymptotic results for certain first-passage times and areas of renewal processes\",\"authors\":\"C. Macci, B. Pacchiarotti\",\"doi\":\"10.1090/tpms/1189\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider the process <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"StartSet x minus upper N left-parenthesis t right-parenthesis colon t greater-than-or-equal-to 0 EndSet\\\"> <mml:semantics> <mml:mrow> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\{x-N(t):t\\\\geq 0\\\\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"x element-of double-struck upper R Subscript plus\\\"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msub> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">x\\\\in \\\\mathbb {R}_+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"StartSet upper N left-parenthesis t right-parenthesis colon t greater-than-or-equal-to 0 EndSet\\\"> <mml:semantics> <mml:mrow> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\{N(t):t\\\\geq 0\\\\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis tau left-parenthesis x right-parenthesis comma upper A left-parenthesis x right-parenthesis right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">(\\\\tau (x),A(x))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"tau left-parenthesis x right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\tau (x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the first-passage time of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"StartSet x minus upper N left-parenthesis t right-parenthesis colon t greater-than-or-equal-to 0 EndSet\\\"> <mml:semantics> <mml:mrow> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\{x-N(t):t\\\\geq 0\\\\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to reach zero or a negative value, and <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A left-parenthesis x right-parenthesis colon-equal integral Subscript 0 Superscript tau left-parenthesis x right-parenthesis Baseline left-parenthesis x minus upper N left-parenthesis t right-parenthesis right-parenthesis d t\\\"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>≔</mml:mo> <mml:msubsup> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mn>0</mml:mn> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">A(x)≔\\\\int _0^{\\\\tau (x)}(x-N(t))dt</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the corresponding first-passage (positive) area swept out by the process <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"StartSet x minus upper N left-parenthesis t right-parenthesis colon t greater-than-or-equal-to 0 EndSet\\\"> <mml:semantics> <mml:mrow> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\{x-N(t):t\\\\geq 0\\\\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We remark that we can define the sequence <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"StartSet left-parenthesis tau left-parenthesis n right-parenthesis comma upper A left-parenthesis n right-parenthesis right-parenthesis colon n greater-than-or-equal-to 1 EndSet\\\"> <mml:semantics> <mml:mrow> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\{(\\\\tau (n),A(n)):n\\\\geq 1\\\\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by referring to the concept of integrated random walk. Our aim is to prove asymptotic results as <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"x right-arrow normal infinity\\\"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo stretchy=\\\"false\\\">→<!-- → --></mml:mo> <mml:mi mathvariant=\\\"normal\\\">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">x\\\\to \\\\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the fashion of large (and moderate) deviations.</p>\",\"PeriodicalId\":42776,\"journal\":{\"name\":\"Theory of Probability and Mathematical Statistics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2021-05-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theory of Probability and Mathematical Statistics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/tpms/1189\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theory of Probability and Mathematical Statistics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/tpms/1189","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
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