Salvato in:
| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2211.14629 |
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Sommario:
- In this paper, we set up the distribution function $$ φ(u)=\mathbb{P}\left(\sup_{n\geqslant 1}\sum_{i=1}^{n}\left(X_i-κ\right)<u\right), $$ and the generating function of $φ(u+1)$, where $u\in\mathbb{N}_0$, $κ\in\mathbb{N}$, the random walk $\left\{\sum_{i=1}^{n}X_i, n\in\mathbb{N}\right\},$ consists of $N\in\mathbb{N}$ periodically occurring distributions, and the integer-valued and non-negative random variables $X_1,\,X_2,\,\ldots$ are independent. This research generalizes two recent works where $\{κ=1,\,N\in\mathbb{N}\}$ and $\{κ\in\mathbb{N},\,N=1\}$ were considered respectively. The provided sequence of sums $\left\{\sum_{i=1}^{n}\left(X_i-κ\right),\,n\in\mathbb{N}\right\}$ generates so-called multi-seasonal discrete-time risk model with arbitrary natural premium and its known distribution enables to calculate the ultimate time ruin probability $1-φ(u)$ or survival probability $φ(u)$. Verifying obtained theoretical statements we demonstrate several computational examples for survival probability $φ(u)$ and its generating function when $\{κ=2,\,N=2\}$, $\{κ=3,\,N=2\}$, $\{κ=5,\,N=10\}$ and $X_i$ admits Poisson and some other distributions. We also conjecture the non-singularity of certain matrices.