An Arc-Sine Law for Last Hitting Points in the Two-Parameter Wiener Space
We develop the two-parameter version of an arc-sine law for a last hitting time. The existing arc-sine laws are about a stochastic process <inline-formula< <math display="inline"< <semantics< <msub< <mi<X</mi< <mi<t</mi< </msub< </semantics< </math< </inline-formula< with one parameter <i<t</i<. If there is another varying key factor of an event described by a process, then we need to consider another parameter besides <i<t</i<. That is, we need a system of random variables with two parameters, say <inline-formula< <math display="inline"< <semantics< <msub< <mi<X</mi< <mrow< <mi<s</mi< <mo<,</mo< <mi<t</mi< </mrow< </msub< </semantics< </math< </inline-formula<, which is far more complex than one-parameter processes. In this paper we challenge to develop such an idea, and provide the two-parameter version of an arc-sine law for a last hitting time. An arc-sine law for a two-parameter process is hardly found in literature. We use the properties of the two-parameter Wiener process for our development. Our result shows that the probability of last hitting points in the two-parameter Wiener space turns out to be arcsine-distributed. One can use our results to predict an event happened in a system of random variables with two parameters, which is not available among existing arc-sine laws for one parameter processes..
| Medienart: |
E-Artikel |
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| Erscheinungsjahr: |
2019 |
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| Erschienen: |
2019 |
| Enthalten in: |
Zur Gesamtaufnahme - volume:7 |
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| Enthalten in: |
Mathematics - 7(2019), 11, p 1131 |
| Sprache: |
Englisch |
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| Beteiligte Personen: |
Jeong-Gyoo KIM [VerfasserIn] |
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| Links: |
doi.org [kostenfrei] |
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| Themen: |
Arc-sine law |
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| doi: |
10.3390/math7111131 |
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| funding: |
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| Förderinstitution / Projekttitel: |
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| PPN (Katalog-ID): |
DOAJ046217010 |
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| 520 | |a We develop the two-parameter version of an arc-sine law for a last hitting time. The existing arc-sine laws are about a stochastic process <inline-formula< <math display="inline"< <semantics< <msub< <mi<X</mi< <mi<t</mi< </msub< </semantics< </math< </inline-formula< with one parameter <i<t</i<. If there is another varying key factor of an event described by a process, then we need to consider another parameter besides <i<t</i<. That is, we need a system of random variables with two parameters, say <inline-formula< <math display="inline"< <semantics< <msub< <mi<X</mi< <mrow< <mi<s</mi< <mo<,</mo< <mi<t</mi< </mrow< </msub< </semantics< </math< </inline-formula<, which is far more complex than one-parameter processes. In this paper we challenge to develop such an idea, and provide the two-parameter version of an arc-sine law for a last hitting time. An arc-sine law for a two-parameter process is hardly found in literature. We use the properties of the two-parameter Wiener process for our development. Our result shows that the probability of last hitting points in the two-parameter Wiener space turns out to be arcsine-distributed. One can use our results to predict an event happened in a system of random variables with two parameters, which is not available among existing arc-sine laws for one parameter processes. | ||
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