# Ruin Probabilities with Dependence on the Number of Claims within a Fixed Time Window

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## Abstract

**:**

## 1. Introduction

## 2. The Model

**Definition 1.**

## 3. Asymptotic Results

#### 3.1. The Heavy-Tailed Case

**Theorem 2.**

**Proof.**

**Remark 1.**

#### 3.2. The Intermediate Case

**Theorem 3.**

**Proof.**

#### 3.3. The Cramér Case

**Theorem 4.**

**Proof.**

**Remark 2.**

**Example 1.**

## 4. Numerical Results

#### 4.1. Importance Sampling and Change of Measure

**Theorem 5.**

**Proof.**

#### 4.2. Embedded Markov Additive Process

**Lemma 6.**

**Proof.**

**Lemma 7.**

**Proof.**

**Lemma 8.**

**Proof.**

**Corollary 9.**

**Theorem 10.**

**Proof.**

**Remark 3.**

**Proof.**

**Example 2.**

#### 4.3. Case Study

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 6.**Examples: ruin probabilities for exponential claims. (

**a**) Ruin probabilities when ${\lambda}_{1}=0.45$, ${\lambda}_{2}=0.15$, $\beta =0.5$; (

**b**) ruin probabilities when ${\lambda}_{1}=0.15,{\lambda}_{2}=0.45,\beta =0.5$.

**Figure 7.**Examples: ruin probabilities for Pareto claims. (

**a**) Ruin probabilities when ${\lambda}_{1}=0.45$, ${\lambda}_{2}=0.15$, $\alpha =2$; (

**b**) ruin probabilities when ${\lambda}_{1}=0.15,{\lambda}_{2}=0.45,\alpha =2$.

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**MDPI and ACS Style**

Constantinescu, C.; Dai, S.; Ni, W.; Palmowski, Z.
Ruin Probabilities with Dependence on the Number of Claims within a Fixed Time Window. *Risks* **2016**, *4*, 17.
https://doi.org/10.3390/risks4020017

**AMA Style**

Constantinescu C, Dai S, Ni W, Palmowski Z.
Ruin Probabilities with Dependence on the Number of Claims within a Fixed Time Window. *Risks*. 2016; 4(2):17.
https://doi.org/10.3390/risks4020017

**Chicago/Turabian Style**

Constantinescu, Corina, Suhang Dai, Weihong Ni, and Zbigniew Palmowski.
2016. "Ruin Probabilities with Dependence on the Number of Claims within a Fixed Time Window" *Risks* 4, no. 2: 17.
https://doi.org/10.3390/risks4020017