{"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}
引用次数: 0
Abstract
Let X\mathbb {X} and Y\mathbb {Y} be two complete, separable, metric spaces, ξε(x),x∈X\xi _\varepsilon (x), x \in \mathbb {X} and νε\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 ξε(νε)\xi _\varepsilon (\nu _\varepsilon ) that is the conditions insuring holding of relation, ξε(νε)⟶dξ0(ν0)\xi _\varepsilon (\nu _\varepsilon ) \stackrel {\mathsf {d}}{\longrightarrow } \xi _0(\nu _0) as ε→0\varepsilon \to 0.