# Ruin Probabilities in a Dependent Discrete-Time Risk Model With Gamma-Like Tailed Insurance Risks

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Main Results

**Theorem**

**1.**

- (1)
- If $0<{y}_{*}<1$, then for each fixed $n\ge 1$, it holds that $\mathbb{E}\left({e}^{\gamma {S}_{n-1}}\right)<\infty ,\phantom{\rule{4pt}{0ex}}\mathbb{E}\left({e}^{\gamma {M}_{n-1}}\right)<\infty $, and$$\begin{array}{c}\mathbb{P}({S}_{n}>x)\sim \frac{{p}_{*}h\left({y}_{*}\right)\mathbb{E}\left({e}^{\gamma {S}_{n-1}}\right)}{{y}_{*}^{\alpha -1}}l\left(x\right){x}^{\alpha -1}{e}^{-\frac{\gamma x}{{y}_{*}}},\hfill \end{array}$$$$\begin{array}{c}\mathbb{P}({M}_{n}>x)\sim \frac{{p}_{*}h\left({y}_{*}\right)\mathbb{E}\left({e}^{\gamma {M}_{n-1}}\right)}{{y}_{*}^{\alpha -1}}l\left(x\right){x}^{\alpha -1}{e}^{-\frac{\gamma x}{{y}_{*}}}.\hfill \end{array}$$
- (2)
- If ${y}_{*}=1$, then for each fixed $n\ge 1$, it holds that$$\mathbb{P}({M}_{n}>x)\sim \mathbb{P}({S}_{n}>x)\sim \frac{{p}_{*}^{n}h\left({y}_{*}\right){\gamma}^{n-1}{(\Gamma \left(\alpha \right))}^{n}}{{y}_{*}^{(\alpha -1)n}\Gamma \left(n\alpha \right)}{\left(l\left(x\right)\right)}^{n}{x}^{n\alpha -1}{e}^{-\gamma x}.$$

**Theorem**

**2.**

## 3. Proofs of Main Results

**Lemma**

**1.**

**Proof**

**of Lemma 1.**

**Proof**

**of Theorem 1.**

**Remark**

**1.**

**Proof**

**of Theorem 2.**

## 4. Simulation Study

Y | 0.2 | 0.6 | 1 |

$\mathbb{P}(Y=y)$ | 0.3 | 0.4 | 0.3 |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Table 1.**Comparison between the simulated estimate values and the asymptotic values in Theorems 1-(2) for $N=1.0\times {10}^{7}$.

x | Simulated Estimate Values | Asymptotic Values |
---|---|---|

100 | $2.06\times {10}^{-5}$ ($1.20451\times {10}^{-6}$) | $1.18\times {10}^{-5}$ |

110 | $1.16\times {10}^{-5}$ ($3.06396\times {10}^{-7}$) | $8.47\times {10}^{-6}$ |

120 | $4.85\times {10}^{-6}$ ($2.51962\times {10}^{-7}$) | $5.73\times {10}^{-6}$ |

130 | $2.05\times {10}^{-6}$ ($1.50271\times {10}^{-7}$) | $3.69\times {10}^{-6}$ |

140 | $1.08\times {10}^{-6}$ ($1.28344\times {10}^{-7}$) | $2.28\times {10}^{-6}$ |

150 | $3.92\times {10}^{-7}$ ($2.11470\times {10}^{-7}$) | $1.36\times {10}^{-6}$ |

**Table 2.**Comparison between the simulated estimate values and the asymptotic values in Theorem 1-(2) for $N=1.5\times {10}^{7}$.

x | Simulated Estimate Values | Asymptotic Values |
---|---|---|

100 | $2.03\times {10}^{-5}$ ($6.75224\times {10}^{-7}$) | $1.18\times {10}^{-5}$ |

110 | $1.23\times {10}^{-5}$ ($3.35691\times {10}^{-7}$) | $8.47\times {10}^{-6}$ |

120 | $5.04\times {10}^{-6}$ ($2.35759\times {10}^{-7}$) | $5.73\times {10}^{-6}$ |

130 | $2.27\times {10}^{-6}$ ($1.04678\times {10}^{-7}$) | $3.69\times {10}^{-6}$ |

140 | $1.45\times {10}^{-6}$ ($6.50245\times {10}^{-8}$) | $2.28\times {10}^{-6}$ |

150 | $6.98\times {10}^{-7}$ ($3.32866\times {10}^{-8}$) | $1.36\times {10}^{-6}$ |

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**MDPI and ACS Style**

Huang, X.-F.; Zhang, T.; Yang, Y.; Jiang, T.
Ruin Probabilities in a Dependent Discrete-Time Risk Model With Gamma-Like Tailed Insurance Risks. *Risks* **2017**, *5*, 14.
https://doi.org/10.3390/risks5010014

**AMA Style**

Huang X-F, Zhang T, Yang Y, Jiang T.
Ruin Probabilities in a Dependent Discrete-Time Risk Model With Gamma-Like Tailed Insurance Risks. *Risks*. 2017; 5(1):14.
https://doi.org/10.3390/risks5010014

**Chicago/Turabian Style**

Huang, Xing-Fang, Ting Zhang, Yang Yang, and Tao Jiang.
2017. "Ruin Probabilities in a Dependent Discrete-Time Risk Model With Gamma-Like Tailed Insurance Risks" *Risks* 5, no. 1: 14.
https://doi.org/10.3390/risks5010014