R. Leipus, Vytaute Pilipauskaite, D. Surgailis
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{"title":"网络流量聚合与随机场的各向异性缩放","authors":"R. Leipus, Vytaute Pilipauskaite, D. Surgailis","doi":"10.1090/tpms/1188","DOIUrl":null,"url":null,"abstract":"<p>We discuss joint spatial-temporal scaling limits of sums <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Subscript lamda comma gamma\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>A</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>λ<!-- λ --></mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>γ<!-- γ --></mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">A_{\\lambda ,\\gamma }</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> (indexed by <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis x comma y right-parenthesis element-of double-struck upper R Subscript plus Superscript 2\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>y</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:msubsup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:mo>+</mml:mo>\n <mml:mn>2</mml:mn>\n </mml:msubsup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(x,y) \\in \\mathbb {R}^2_+</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>) of large number <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis lamda Superscript gamma Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>O</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>λ<!-- λ --></mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>γ<!-- γ --></mml:mi>\n </mml:mrow>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">O(\\lambda ^{\\gamma })</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of independent copies of integrated input process <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X equals StartSet upper X left-parenthesis t right-parenthesis comma t element-of double-struck upper R EndSet\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>X</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\n <mml:mi>X</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>t</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mi>t</mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">X = \\{X(t), t \\in \\mathbb {R}\\}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> at time scale <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda\">\n <mml:semantics>\n <mml:mi>λ<!-- λ --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\lambda</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, for any given <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"gamma greater-than 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>γ<!-- γ --></mml:mi>\n <mml:mo>></mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\gamma >0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We consider two classes of inputs <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>: (I) Poisson shot-noise with (random) pulse process, and (II) regenerative process with random pulse process and regeneration times following a heavy-tailed stationary renewal process. The above classes include several queueing and network traffic models for which joint spatial-temporal limits were previously discussed in the literature. In both cases (I) and (II) we find simple conditions on the input process in order that the normalized random fields <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Subscript lamda comma gamma\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>A</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>λ<!-- λ --></mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>γ<!-- γ --></mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">A_{\\lambda ,\\gamma }</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> tend to an <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"alpha\">\n <mml:semantics>\n <mml:mi>α<!-- α --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\alpha</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-stable Lévy sheet <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis 1 greater-than alpha greater-than 2 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo>></mml:mo>\n <mml:mi>α<!-- α --></mml:mi>\n <mml:mo>></mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(1> \\alpha >2)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"gamma greater-than gamma 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>γ<!-- γ --></mml:mi>\n <mml:mo>></mml:mo>\n <mml:msub>\n <mml:mi>γ<!-- γ --></mml:mi>\n <mml:mn>0</mml:mn>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\gamma > \\gamma _0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, and to a fractional Brownian sheet if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"gamma greater-than gamma 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>γ<!-- γ --></mml:mi>\n <mml:mo>></mml:mo>\n <mml:msub>\n <mml:mi>γ<!-- γ --></mml:mi>\n <mml:mn>0</mml:mn>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\gamma > \\gamma _0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, for some <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"gamma 0 greater-than 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>γ<!-- γ --></mml:mi>\n <mml:mn>0</mml:mn>\n </mml:msub>\n <mml:mo>></mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\gamma _0>0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We also prove an ‘intermediate’ limit for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"gamma equals gamma 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>γ<!-- γ --></mml:mi>\n <mml:mo>=</mml:mo>\n <mml:msub>\n <mml:mi>γ<!-- γ --></mml:mi>\n <mml:mn>0</mml:mn>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\gamma = \\gamma _0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Our results extend the previous works of R. Gaigalas and I. Kaj [Bernoulli 9 (2003), no. 4, 671–703] and T. Mikosch, S. Resnick, H. Rootzén and A. Stegeman [Ann. Appl. Probab. 12 (2002), no. 1, 23–68] and other papers to more general and new input processes.</p>","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2021-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Aggregation of network traffic and anisotropic scaling of random fields\",\"authors\":\"R. Leipus, Vytaute Pilipauskaite, D. Surgailis\",\"doi\":\"10.1090/tpms/1188\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We discuss joint spatial-temporal scaling limits of sums <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A Subscript lamda comma gamma\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi>A</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>λ<!-- λ --></mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>γ<!-- γ --></mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">A_{\\\\lambda ,\\\\gamma }</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> (indexed by <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis x comma y right-parenthesis element-of double-struck upper R Subscript plus Superscript 2\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>y</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>∈<!-- ∈ --></mml:mo>\\n <mml:msubsup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n <mml:mo>+</mml:mo>\\n <mml:mn>2</mml:mn>\\n </mml:msubsup>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">(x,y) \\\\in \\\\mathbb {R}^2_+</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>) of large number <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper O left-parenthesis lamda Superscript gamma Baseline right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>O</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:msup>\\n <mml:mi>λ<!-- λ --></mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>γ<!-- γ --></mml:mi>\\n </mml:mrow>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">O(\\\\lambda ^{\\\\gamma })</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> of independent copies of integrated input process <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper X equals StartSet upper X left-parenthesis t right-parenthesis comma t element-of double-struck upper R EndSet\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>X</mml:mi>\\n <mml:mo>=</mml:mo>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo>\\n <mml:mi>X</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>t</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>,</mml:mo>\\n <mml:mi>t</mml:mi>\\n <mml:mo>∈<!-- ∈ --></mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">X = \\\\{X(t), t \\\\in \\\\mathbb {R}\\\\}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> at time scale <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"lamda\\\">\\n <mml:semantics>\\n <mml:mi>λ<!-- λ --></mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\lambda</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, for any given <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"gamma greater-than 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>γ<!-- γ --></mml:mi>\\n <mml:mo>></mml:mo>\\n <mml:mn>0</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\gamma >0</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. We consider two classes of inputs <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper X\\\">\\n <mml:semantics>\\n <mml:mi>X</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">X</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>: (I) Poisson shot-noise with (random) pulse process, and (II) regenerative process with random pulse process and regeneration times following a heavy-tailed stationary renewal process. The above classes include several queueing and network traffic models for which joint spatial-temporal limits were previously discussed in the literature. In both cases (I) and (II) we find simple conditions on the input process in order that the normalized random fields <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A Subscript lamda comma gamma\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi>A</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>λ<!-- λ --></mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>γ<!-- γ --></mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">A_{\\\\lambda ,\\\\gamma }</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> tend to an <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"alpha\\\">\\n <mml:semantics>\\n <mml:mi>α<!-- α --></mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\alpha</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-stable Lévy sheet <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis 1 greater-than alpha greater-than 2 right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mo>></mml:mo>\\n <mml:mi>α<!-- α --></mml:mi>\\n <mml:mo>></mml:mo>\\n <mml:mn>2</mml:mn>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">(1> \\\\alpha >2)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> if <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"gamma greater-than gamma 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>γ<!-- γ --></mml:mi>\\n <mml:mo>></mml:mo>\\n <mml:msub>\\n <mml:mi>γ<!-- γ --></mml:mi>\\n <mml:mn>0</mml:mn>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\gamma > \\\\gamma _0</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, and to a fractional Brownian sheet if <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"gamma greater-than gamma 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>γ<!-- γ --></mml:mi>\\n <mml:mo>></mml:mo>\\n <mml:msub>\\n <mml:mi>γ<!-- γ --></mml:mi>\\n <mml:mn>0</mml:mn>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\gamma > \\\\gamma _0</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, for some <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"gamma 0 greater-than 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>γ<!-- γ --></mml:mi>\\n <mml:mn>0</mml:mn>\\n </mml:msub>\\n <mml:mo>></mml:mo>\\n <mml:mn>0</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\gamma _0>0</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. We also prove an ‘intermediate’ limit for <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"gamma equals gamma 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>γ<!-- γ --></mml:mi>\\n <mml:mo>=</mml:mo>\\n <mml:msub>\\n <mml:mi>γ<!-- γ --></mml:mi>\\n <mml:mn>0</mml:mn>\\n </mml:msub>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\gamma = \\\\gamma _0</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. Our results extend the previous works of R. Gaigalas and I. Kaj [Bernoulli 9 (2003), no. 4, 671–703] and T. Mikosch, S. Resnick, H. Rootzén and A. Stegeman [Ann. Appl. Probab. 12 (2002), no. 1, 23–68] and other papers to more general and new input processes.</p>\",\"PeriodicalId\":42776,\"journal\":{\"name\":\"Theory of Probability and Mathematical Statistics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2021-12-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theory of Probability and Mathematical Statistics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/tpms/1188\",\"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/1188","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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