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A problem of Foldes and Puri on the Wiener process
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| Zeitschriftentitel: | Transactions of the American Mathematical Society |
|---|---|
| Personen und Körperschaften: | |
| In: | Transactions of the American Mathematical Society, 348, 1996, 1, S. 219-228 |
| Format: | E-Article |
| Sprache: | Englisch |
| veröffentlicht: |
American Mathematical Society (AMS)
|
| Schlagwörter: |
| author_facet |
Shi, Z. Shi, Z. |
|---|---|
| author |
Shi, Z. |
| spellingShingle |
Shi, Z. Transactions of the American Mathematical Society A problem of Foldes and Puri on the Wiener process Applied Mathematics General Mathematics |
| author_sort |
shi, z. |
| spelling |
Shi, Z. 0002-9947 1088-6850 American Mathematical Society (AMS) Applied Mathematics General Mathematics http://dx.doi.org/10.1090/s0002-9947-96-01485-7 <p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a real-valued Wiener process starting from 0, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\tau (t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the right-continuous inverse process of its local time at 0. Földes and Puri [3] raise the problem of studying the almost sure asymptotic behavior of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X left-parenthesis t right-parenthesis equals integral Subscript 0 Superscript tau left-parenthesis t right-parenthesis Baseline double-struck 1 Subscript left-brace StartAbsoluteValue upper W left-parenthesis u right-parenthesis EndAbsoluteValue less-than-or-equal-to alpha t right-brace Baseline d u"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</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>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn mathvariant="double-struck">1</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>W</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mi>t</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> </mml:msub> <mml:mspace width="thinmathspace" /> <mml:mi>d</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X(t)=\int _0^{\tau (t)} \mathbb {1}_{\{|W(u)| \le \alpha t\}}\,du</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> tends to infinity, i.e. they ask: how long does <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> stay in a tube before “crossing very much” a given level? In this note, both limsup and liminf laws of the iterated logarithm are provided for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p> A problem of Foldes and Puri on the Wiener process Transactions of the American Mathematical Society |
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10.1090/s0002-9947-96-01485-7 |
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Online Free |
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Mathematik |
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ElectronicArticle |
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ai-49-aHR0cDovL2R4LmRvaS5vcmcvMTAuMTA5MC9zMDAwMi05OTQ3LTk2LTAxNDg1LTc |
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DE-Ch1 DE-L229 DE-D275 DE-Bn3 DE-Brt1 DE-D161 DE-Zwi2 DE-Gla1 DE-Zi4 DE-15 DE-Pl11 DE-Rs1 DE-105 DE-14 |
| imprint |
American Mathematical Society (AMS), 1996 |
| imprint_str_mv |
American Mathematical Society (AMS), 1996 |
| issn |
0002-9947 1088-6850 |
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0002-9947 1088-6850 |
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English |
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American Mathematical Society (AMS) (CrossRef) |
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shi1996aproblemoffoldesandpurionthewienerprocess |
| publishDateSort |
1996 |
| publisher |
American Mathematical Society (AMS) |
| recordtype |
ai |
| record_format |
ai |
| series |
Transactions of the American Mathematical Society |
| source_id |
49 |
| title |
A problem of Foldes and Puri on the Wiener process |
| title_unstemmed |
A problem of Foldes and Puri on the Wiener process |
| title_full |
A problem of Foldes and Puri on the Wiener process |
| title_fullStr |
A problem of Foldes and Puri on the Wiener process |
| title_full_unstemmed |
A problem of Foldes and Puri on the Wiener process |
| title_short |
A problem of Foldes and Puri on the Wiener process |
| title_sort |
a problem of foldes and puri on the wiener process |
| topic |
Applied Mathematics General Mathematics |
| url |
http://dx.doi.org/10.1090/s0002-9947-96-01485-7 |
| publishDate |
1996 |
| physical |
219-228 |
| description |
<p>Let <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W">
<mml:semantics>
<mml:mi>W</mml:mi>
<mml:annotation encoding="application/x-tex">W</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> be a real-valued Wiener process starting from 0, and <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau left-parenthesis t right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mi>τ<!-- τ --></mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>t</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\tau (t)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> be the right-continuous inverse process of its local time at 0. Földes and Puri [3] raise the problem of studying the almost sure asymptotic behavior of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X left-parenthesis t right-parenthesis equals integral Subscript 0 Superscript tau left-parenthesis t right-parenthesis Baseline double-struck 1 Subscript left-brace StartAbsoluteValue upper W left-parenthesis u right-parenthesis EndAbsoluteValue less-than-or-equal-to alpha t right-brace Baseline d u">
<mml:semantics>
<mml:mrow>
<mml:mi>X</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>t</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>t</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:msubsup>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mn mathvariant="double-struck">1</mml:mn>
</mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo fence="false" stretchy="false">{</mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">|</mml:mo>
</mml:mrow>
<mml:mi>W</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>u</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">|</mml:mo>
</mml:mrow>
<mml:mo>≤<!-- ≤ --></mml:mo>
<mml:mi>α<!-- α --></mml:mi>
<mml:mi>t</mml:mi>
<mml:mo fence="false" stretchy="false">}</mml:mo>
</mml:mrow>
</mml:msub>
<mml:mspace width="thinmathspace" />
<mml:mi>d</mml:mi>
<mml:mi>u</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">X(t)=\int _0^{\tau (t)} \mathbb {1}_{\{|W(u)| \le \alpha t\}}\,du</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> as <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t">
<mml:semantics>
<mml:mi>t</mml:mi>
<mml:annotation encoding="application/x-tex">t</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> tends to infinity, i.e. they ask: how long does <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W">
<mml:semantics>
<mml:mi>W</mml:mi>
<mml:annotation encoding="application/x-tex">W</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> stay in a tube before “crossing very much” a given level? In this note, both limsup and liminf laws of the iterated logarithm are provided for <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X">
<mml:semantics>
<mml:mi>X</mml:mi>
<mml:annotation encoding="application/x-tex">X</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>.</p> |
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| description | <p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a real-valued Wiener process starting from 0, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\tau (t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the right-continuous inverse process of its local time at 0. Földes and Puri [3] raise the problem of studying the almost sure asymptotic behavior of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X left-parenthesis t right-parenthesis equals integral Subscript 0 Superscript tau left-parenthesis t right-parenthesis Baseline double-struck 1 Subscript left-brace StartAbsoluteValue upper W left-parenthesis u right-parenthesis EndAbsoluteValue less-than-or-equal-to alpha t right-brace Baseline d u"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</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>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn mathvariant="double-struck">1</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>W</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mi>t</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> </mml:msub> <mml:mspace width="thinmathspace" /> <mml:mi>d</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X(t)=\int _0^{\tau (t)} \mathbb {1}_{\{|W(u)| \le \alpha t\}}\,du</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> tends to infinity, i.e. they ask: how long does <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> stay in a tube before “crossing very much” a given level? In this note, both limsup and liminf laws of the iterated logarithm are provided for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p> |
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| id | ai-49-aHR0cDovL2R4LmRvaS5vcmcvMTAuMTA5MC9zMDAwMi05OTQ3LTk2LTAxNDg1LTc |
| imprint | American Mathematical Society (AMS), 1996 |
| imprint_str_mv | American Mathematical Society (AMS), 1996 |
| institution | DE-Ch1, DE-L229, DE-D275, DE-Bn3, DE-Brt1, DE-D161, DE-Zwi2, DE-Gla1, DE-Zi4, DE-15, DE-Pl11, DE-Rs1, DE-105, DE-14 |
| issn | 0002-9947, 1088-6850 |
| issn_str_mv | 0002-9947, 1088-6850 |
| language | English |
| last_indexed | 2024-03-01T14:19:17.93Z |
| match_str | shi1996aproblemoffoldesandpurionthewienerprocess |
| mega_collection | American Mathematical Society (AMS) (CrossRef) |
| physical | 219-228 |
| publishDate | 1996 |
| publishDateSort | 1996 |
| publisher | American Mathematical Society (AMS) |
| record_format | ai |
| recordtype | ai |
| series | Transactions of the American Mathematical Society |
| source_id | 49 |
| spelling | Shi, Z. 0002-9947 1088-6850 American Mathematical Society (AMS) Applied Mathematics General Mathematics http://dx.doi.org/10.1090/s0002-9947-96-01485-7 <p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a real-valued Wiener process starting from 0, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\tau (t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the right-continuous inverse process of its local time at 0. Földes and Puri [3] raise the problem of studying the almost sure asymptotic behavior of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X left-parenthesis t right-parenthesis equals integral Subscript 0 Superscript tau left-parenthesis t right-parenthesis Baseline double-struck 1 Subscript left-brace StartAbsoluteValue upper W left-parenthesis u right-parenthesis EndAbsoluteValue less-than-or-equal-to alpha t right-brace Baseline d u"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</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>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn mathvariant="double-struck">1</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>W</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mi>t</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> </mml:msub> <mml:mspace width="thinmathspace" /> <mml:mi>d</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X(t)=\int _0^{\tau (t)} \mathbb {1}_{\{|W(u)| \le \alpha t\}}\,du</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> tends to infinity, i.e. they ask: how long does <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> stay in a tube before “crossing very much” a given level? In this note, both limsup and liminf laws of the iterated logarithm are provided for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.</p> A problem of Foldes and Puri on the Wiener process Transactions of the American Mathematical Society |
| spellingShingle | Shi, Z., Transactions of the American Mathematical Society, A problem of Foldes and Puri on the Wiener process, Applied Mathematics, General Mathematics |
| title | A problem of Foldes and Puri on the Wiener process |
| title_full | A problem of Foldes and Puri on the Wiener process |
| title_fullStr | A problem of Foldes and Puri on the Wiener process |
| title_full_unstemmed | A problem of Foldes and Puri on the Wiener process |
| title_short | A problem of Foldes and Puri on the Wiener process |
| title_sort | a problem of foldes and puri on the wiener process |
| title_unstemmed | A problem of Foldes and Puri on the Wiener process |
| topic | Applied Mathematics, General Mathematics |
| url | http://dx.doi.org/10.1090/s0002-9947-96-01485-7 |