On the exact survival probability by setting discrete random variables in E. Sparre Andersen's model
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917558768107520 |
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| author | Grigutis, Andrius |
| author_facet | Grigutis, Andrius |
| contents | In this work, we propose a simplification of the Pollaczek-Khinchine formula for the ultimate time survival (or ruin) probability calculation in exchange for a few assumptions on the random variables which generate the renewal risk model. More precisely, we show the expressibility of the distribution function $$ \mathbb{P}\left(\sup_{n\geqslant1}\sum_{i=1}^{n}(X_i-cθ_i)<u\right),\,u\in\mathbb{N}_0 $$ via the roots of the probability generating function $G_{X-cθ}(s)=1$, the expectation $\mathbb{E}(X-cθ)$, and the probability mass function of $X-cθ$. We assume that the random variables $X_1,\,X_2,\,\ldots$ and $cθ_1,\,cθ_2,\,\ldots$ are independent copies of $X$ and $cθ$ respectively, $c>0$, $X$ and $cθ$ are independent non-negative and integer-valued, and the support of $θ$ is finite. We give few numerical outputs of the proven theoretical statements when the mentioned random variables admit some particular distributions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_16897 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the exact survival probability by setting discrete random variables in E. Sparre Andersen's model Grigutis, Andrius Probability 60G50, 60J80, 91G05 In this work, we propose a simplification of the Pollaczek-Khinchine formula for the ultimate time survival (or ruin) probability calculation in exchange for a few assumptions on the random variables which generate the renewal risk model. More precisely, we show the expressibility of the distribution function $$ \mathbb{P}\left(\sup_{n\geqslant1}\sum_{i=1}^{n}(X_i-cθ_i)<u\right),\,u\in\mathbb{N}_0 $$ via the roots of the probability generating function $G_{X-cθ}(s)=1$, the expectation $\mathbb{E}(X-cθ)$, and the probability mass function of $X-cθ$. We assume that the random variables $X_1,\,X_2,\,\ldots$ and $cθ_1,\,cθ_2,\,\ldots$ are independent copies of $X$ and $cθ$ respectively, $c>0$, $X$ and $cθ$ are independent non-negative and integer-valued, and the support of $θ$ is finite. We give few numerical outputs of the proven theoretical statements when the mentioned random variables admit some particular distributions. |
| title | On the exact survival probability by setting discrete random variables in E. Sparre Andersen's model |
| topic | Probability 60G50, 60J80, 91G05 |
| url | https://arxiv.org/abs/2306.16897 |