Product Convolution of Generalized Subexponential Distributions
Assume that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathM...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/1/248 |
| _version_ | 1797440408622465024 |
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| author | Gustas Mikutavičius Jonas Šiaulys |
| author_facet | Gustas Mikutavičius Jonas Šiaulys |
| author_sort | Gustas Mikutavičius |
| collection | DOAJ |
| description | Assume that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula> are two independent random variables with distribution functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>η</mi></msub></semantics></math></inline-formula>, respectively. The distribution of a random variable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ξ</mi><mi>η</mi></mrow></semantics></math></inline-formula>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mi>ξ</mi></msub><mo>⊗</mo><msub><mi>F</mi><mi>η</mi></msub></mrow></semantics></math></inline-formula>, is called the product-convolution of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>η</mi></msub></semantics></math></inline-formula>. It is proved that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mi>ξ</mi></msub><mo>⊗</mo><msub><mi>F</mi><mi>η</mi></msub></mrow></semantics></math></inline-formula> is a generalized subexponential distribution if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula> belongs to the class of generalized subexponential distributions and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula> is nonnegative and not degenerated at zero. |
| first_indexed | 2024-03-09T12:07:43Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T12:07:43Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-7a76162c5aa840509e2d13b5bed8cf3b2023-11-30T22:55:53ZengMDPI AGMathematics2227-73902023-01-0111124810.3390/math11010248Product Convolution of Generalized Subexponential DistributionsGustas Mikutavičius0Jonas Šiaulys1Institute of Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, LithuaniaInstitute of Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, LithuaniaAssume that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula> are two independent random variables with distribution functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>η</mi></msub></semantics></math></inline-formula>, respectively. The distribution of a random variable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ξ</mi><mi>η</mi></mrow></semantics></math></inline-formula>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mi>ξ</mi></msub><mo>⊗</mo><msub><mi>F</mi><mi>η</mi></msub></mrow></semantics></math></inline-formula>, is called the product-convolution of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>η</mi></msub></semantics></math></inline-formula>. It is proved that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mi>ξ</mi></msub><mo>⊗</mo><msub><mi>F</mi><mi>η</mi></msub></mrow></semantics></math></inline-formula> is a generalized subexponential distribution if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula> belongs to the class of generalized subexponential distributions and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula> is nonnegative and not degenerated at zero.https://www.mdpi.com/2227-7390/11/1/248tail functionclosure propertyproduct-convolutiongeneralized subexponential distributionheavy-tailed distribution |
| spellingShingle | Gustas Mikutavičius Jonas Šiaulys Product Convolution of Generalized Subexponential Distributions Mathematics tail function closure property product-convolution generalized subexponential distribution heavy-tailed distribution |
| title | Product Convolution of Generalized Subexponential Distributions |
| title_full | Product Convolution of Generalized Subexponential Distributions |
| title_fullStr | Product Convolution of Generalized Subexponential Distributions |
| title_full_unstemmed | Product Convolution of Generalized Subexponential Distributions |
| title_short | Product Convolution of Generalized Subexponential Distributions |
| title_sort | product convolution of generalized subexponential distributions |
| topic | tail function closure property product-convolution generalized subexponential distribution heavy-tailed distribution |
| url | https://www.mdpi.com/2227-7390/11/1/248 |
| work_keys_str_mv | AT gustasmikutavicius productconvolutionofgeneralizedsubexponentialdistributions AT jonassiaulys productconvolutionofgeneralizedsubexponentialdistributions |