A functional limit theorem for random processes with immigration in the case of heavy tails
Let $(X_{k},\xi _{k})_{k\in \mathbb{N}}$ be a sequence of independent copies of a pair $(X,\xi )$ where X is a random process with paths in the Skorokhod space $D[0,\infty )$ and ξ is a positive random variable. The random process with immigration $(Y(u))_{u\in \mathbb{R}}$ is defined as the a.s. fi...
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doaj-b0dd3499de0e450588d547435e973e6e2020-11-24T21:53:22ZengVTeXModern Stochastics: Theory and Applications2351-60462351-60542017-04-01429310810.15559/17-VMSTA76A functional limit theorem for random processes with immigration in the case of heavy tailsAlexander Marynych0Glib Verovkin1Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, UkraineFaculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, UkraineLet $(X_{k},\xi _{k})_{k\in \mathbb{N}}$ be a sequence of independent copies of a pair $(X,\xi )$ where X is a random process with paths in the Skorokhod space $D[0,\infty )$ and ξ is a positive random variable. The random process with immigration $(Y(u))_{u\in \mathbb{R}}$ is defined as the a.s. finite sum $Y(u)=\sum _{k\ge 0}X_{k+1}(u-\xi _{1}-\cdots -\xi _{k})\mathbb{1}_{\{\xi _{1}+\cdots +\xi _{k}\le u\}}$. We obtain a functional limit theorem for the process $(Y(ut))_{u\ge 0}$, as $t\to \infty $, when the law of ξ belongs to the domain of attraction of an α-stable law with $\alpha \in (0,1)$, and the process X oscillates moderately around its mean $\mathbb{E}[X(t)]$. In this situation the process $(Y(ut))_{u\ge 0}$, when scaled appropriately, converges weakly in the Skorokhod space $D(0,\infty )$ to a fractionally integrated inverse stable subordinator.https://vmsta.vtex.vmt/doi/10.15559/17-VMSTA76Fractionally integrated inverse stable subordinatorsrandom process with immigrationshot noise process |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Alexander Marynych Glib Verovkin |
spellingShingle |
Alexander Marynych Glib Verovkin A functional limit theorem for random processes with immigration in the case of heavy tails Modern Stochastics: Theory and Applications Fractionally integrated inverse stable subordinators random process with immigration shot noise process |
author_facet |
Alexander Marynych Glib Verovkin |
author_sort |
Alexander Marynych |
title |
A functional limit theorem for random processes with immigration in the case of heavy tails |
title_short |
A functional limit theorem for random processes with immigration in the case of heavy tails |
title_full |
A functional limit theorem for random processes with immigration in the case of heavy tails |
title_fullStr |
A functional limit theorem for random processes with immigration in the case of heavy tails |
title_full_unstemmed |
A functional limit theorem for random processes with immigration in the case of heavy tails |
title_sort |
functional limit theorem for random processes with immigration in the case of heavy tails |
publisher |
VTeX |
series |
Modern Stochastics: Theory and Applications |
issn |
2351-6046 2351-6054 |
publishDate |
2017-04-01 |
description |
Let $(X_{k},\xi _{k})_{k\in \mathbb{N}}$ be a sequence of independent copies of a pair $(X,\xi )$ where X is a random process with paths in the Skorokhod space $D[0,\infty )$ and ξ is a positive random variable. The random process with immigration $(Y(u))_{u\in \mathbb{R}}$ is defined as the a.s. finite sum $Y(u)=\sum _{k\ge 0}X_{k+1}(u-\xi _{1}-\cdots -\xi _{k})\mathbb{1}_{\{\xi _{1}+\cdots +\xi _{k}\le u\}}$. We obtain a functional limit theorem for the process $(Y(ut))_{u\ge 0}$, as $t\to \infty $, when the law of ξ belongs to the domain of attraction of an α-stable law with $\alpha \in (0,1)$, and the process X oscillates moderately around its mean $\mathbb{E}[X(t)]$. In this situation the process $(Y(ut))_{u\ge 0}$, when scaled appropriately, converges weakly in the Skorokhod space $D(0,\infty )$ to a fractionally integrated inverse stable subordinator. |
topic |
Fractionally integrated inverse stable subordinators random process with immigration shot noise process |
url |
https://vmsta.vtex.vmt/doi/10.15559/17-VMSTA76 |
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