随机停止随机场分布的收敛性

IF 0.4 Q4 STATISTICS & PROBABILITY Theory of Probability and Mathematical Statistics Pub Date : 2021-12-07 DOI:10.1090/tpms/1160
D. Silvestrov
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Silvestrov","doi":"10.1090/tpms/1160","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper X\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">X</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {X}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Y\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Y</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {Y}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be two complete, separable, metric spaces, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"xi Subscript epsilon Baseline left-parenthesis x right-parenthesis comma x element-of double-struck upper X\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>ξ<!-- ξ --></mml:mi>\n <mml:mi>ε<!-- ε --></mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">X</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\xi _\\varepsilon (x), x \\in \\mathbb {X}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"nu Subscript epsilon\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>ν<!-- ν --></mml:mi>\n <mml:mi>ε<!-- ε --></mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\nu _\\varepsilon</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be, for every <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon element-of left-bracket 0 comma 1 right-bracket\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>ε<!-- ε --></mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">]</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\varepsilon \\in [0, 1]</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, respectively, a random field taking values in space <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Y\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Y</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {Y}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and a random variable taking values in space <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper X\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">X</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {X}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We present general conditions for convergence in distribution for random variables <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"xi Subscript epsilon Baseline left-parenthesis nu Subscript epsilon Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>ξ<!-- ξ --></mml:mi>\n <mml:mi>ε<!-- ε --></mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>ν<!-- ν --></mml:mi>\n <mml:mi>ε<!-- ε --></mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\xi _\\varepsilon (\\nu _\\varepsilon )</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> that is the conditions insuring holding of relation, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"xi Subscript epsilon Baseline left-parenthesis nu Subscript epsilon Baseline right-parenthesis long right-arrow Overscript sans-serif d Endscripts xi 0 left-parenthesis nu 0 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>ξ<!-- ξ --></mml:mi>\n <mml:mi>ε<!-- ε --></mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>ν<!-- ν --></mml:mi>\n <mml:mi>ε<!-- ε --></mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-REL\">\n <mml:mover>\n <mml:mrow class=\"MJX-TeXAtom-OP MJX-fixedlimits\">\n <mml:mo stretchy=\"false\">⟶<!-- ⟶ --></mml:mo>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"sans-serif\">d</mml:mi>\n </mml:mrow>\n </mml:mrow>\n </mml:mover>\n </mml:mrow>\n <mml:msub>\n <mml:mi>ξ<!-- ξ --></mml:mi>\n <mml:mn>0</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>ν<!-- ν --></mml:mi>\n <mml:mn>0</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\xi _\\varepsilon (\\nu _\\varepsilon ) \\stackrel {\\mathsf {d}}{\\longrightarrow } \\xi _0(\\nu _0)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> as <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon right-arrow 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>ε<!-- ε --></mml:mi>\n <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\varepsilon \\to 0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2021-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Convergence in distribution for randomly stopped random fields\",\"authors\":\"D. Silvestrov\",\"doi\":\"10.1090/tpms/1160\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper X\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">X</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {X}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper Y\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">Y</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {Y}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> be two complete, separable, metric spaces, <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"xi Subscript epsilon Baseline left-parenthesis x right-parenthesis comma x element-of double-struck upper X\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>ξ<!-- ξ --></mml:mi>\\n <mml:mi>ε<!-- ε --></mml:mi>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>,</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo>∈<!-- ∈ --></mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">X</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\xi _\\\\varepsilon (x), x \\\\in \\\\mathbb {X}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"nu Subscript epsilon\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi>ν<!-- ν --></mml:mi>\\n <mml:mi>ε<!-- ε --></mml:mi>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\nu _\\\\varepsilon</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> be, for every <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"epsilon element-of left-bracket 0 comma 1 right-bracket\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>ε<!-- ε --></mml:mi>\\n <mml:mo>∈<!-- ∈ --></mml:mo>\\n <mml:mo stretchy=\\\"false\\\">[</mml:mo>\\n <mml:mn>0</mml:mn>\\n <mml:mo>,</mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mo stretchy=\\\"false\\\">]</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\varepsilon \\\\in [0, 1]</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, respectively, a random field taking values in space <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper Y\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">Y</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {Y}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and a random variable taking values in space <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper X\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">X</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {X}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. We present general conditions for convergence in distribution for random variables <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"xi Subscript epsilon Baseline left-parenthesis nu Subscript epsilon Baseline right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>ξ<!-- ξ --></mml:mi>\\n <mml:mi>ε<!-- ε --></mml:mi>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:msub>\\n <mml:mi>ν<!-- ν --></mml:mi>\\n <mml:mi>ε<!-- ε --></mml:mi>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\xi _\\\\varepsilon (\\\\nu _\\\\varepsilon )</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> that is the conditions insuring holding of relation, <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"xi Subscript epsilon Baseline left-parenthesis nu Subscript epsilon Baseline right-parenthesis long right-arrow Overscript sans-serif d Endscripts xi 0 left-parenthesis nu 0 right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>ξ<!-- ξ --></mml:mi>\\n <mml:mi>ε<!-- ε --></mml:mi>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:msub>\\n <mml:mi>ν<!-- ν --></mml:mi>\\n <mml:mi>ε<!-- ε --></mml:mi>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-REL\\\">\\n <mml:mover>\\n <mml:mrow class=\\\"MJX-TeXAtom-OP MJX-fixedlimits\\\">\\n <mml:mo stretchy=\\\"false\\\">⟶<!-- ⟶ --></mml:mo>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"sans-serif\\\">d</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n </mml:mover>\\n </mml:mrow>\\n <mml:msub>\\n <mml:mi>ξ<!-- ξ --></mml:mi>\\n <mml:mn>0</mml:mn>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:msub>\\n <mml:mi>ν<!-- ν --></mml:mi>\\n <mml:mn>0</mml:mn>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\xi _\\\\varepsilon (\\\\nu _\\\\varepsilon ) \\\\stackrel {\\\\mathsf {d}}{\\\\longrightarrow } \\\\xi _0(\\\\nu _0)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> as <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"epsilon right-arrow 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>ε<!-- ε --></mml:mi>\\n <mml:mo stretchy=\\\"false\\\">→<!-- → --></mml:mo>\\n <mml:mn>0</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\varepsilon \\\\to 0</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>.</p>\",\"PeriodicalId\":42776,\"journal\":{\"name\":\"Theory of Probability and Mathematical Statistics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2021-12-07\",\"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/1160\",\"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/1160","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0

摘要

设X \mathbb X{和Y }\mathbb Y{是两个完备的,可分离的度量空间,ξ ε (X), X∈X }\xi _ \varepsilon (X), X \in\mathbb X{和ν ε }\nu _ \varepsilon be,对于每一个ε∈[0],1] \varepsilon\in[0,1]分别为在空间Y中取值的随机场\mathbb Y{和在空间X中取值的随机变量}\mathbb X{。给出了随机变量ξ ε (ν ε) }\xi _ \varepsilon (\nu _ \varepsilon)分布收敛的一般条件,即保证关系成立的条件。ξ ε (ν ε) δ ξ 0(ν 0) \xi _ \varepsilon (\nu _ \varepsilon) \stackrel{\mathsf d{}}{\longrightarrow}\xi _0(\nu _0)为ε→0\varepsilon\to
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Convergence in distribution for randomly stopped random fields

Let X \mathbb {X} and Y \mathbb {Y} be two complete, separable, metric spaces, ξ ε ( x ) , x X \xi _\varepsilon (x), x \in \mathbb {X} and ν ε \nu _\varepsilon be, for every ε [ 0 , 1 ] \varepsilon \in [0, 1] , respectively, a random field taking values in space Y \mathbb {Y} and a random variable taking values in space X \mathbb {X} . We present general conditions for convergence in distribution for random variables ξ ε ( ν ε ) \xi _\varepsilon (\nu _\varepsilon ) that is the conditions insuring holding of relation, ξ ε ( ν ε ) d ξ 0 ( ν 0 ) \xi _\varepsilon (\nu _\varepsilon ) \stackrel {\mathsf {d}}{\longrightarrow } \xi _0(\nu _0) as ε 0 \varepsilon \to 0 .

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