Precise Large Deviations of Aggregate Claims with Dominated Variation in Dependent Multi-Risk Models

Shijie Wang,Wensheng Wang,Xuejun Wang

doi:10.1155/2014/972029

Shijie Wang, Wensheng Wang + Show 1 more

Open Access

https://doi.org/10.1155/2014/972029

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Abstract

We consider a dependent multirisk model in insurance, where all the claims constitute a linearly extended negatively orthant dependent (LENOD) random array, and then upper and lower bounds for precise large deviations of nonrandom and random sums of random variables with dominated variation are investigated. The obtained results extend some related existing ones.

Highlights

In a classic insurance risk model the surplus is described as the initial surplus plus the premium income with the claims taken off
In the consideration of the need of studying multivariate random variables, we introduce the concept of Linearly Extended Negatively Orthant Dependent (LENOD) random arrays in this paper
Precise large deviations for nonrandom sums and random sums with dominated variation in dependent multi-risk models are presented in Sections 3 and 4

Summary

Introduction

In a classic insurance risk model the surplus is described as the initial surplus plus the premium income with the claims taken off. The relation (2) describes the so-called precise large deviations for random sums in multi-risk models. Abstract and Applied Analysis independent structures, respectively He et al [20] obtained the lower bounds of precise large deviations of multi-risk models with nonnegative random variables (regardless of heavy or light tails) under a specific dependence structure. Motivated by the two reasons mentioned above, in this paper, upper and lower bounds for precise large deviations of aggregate claims with dominated variation in dependent multi-risk models (see the dependent structures in Definitions 1 and 2 below) are investigated. In this paper, we redefine a new dependence structure called LENOD for random arrays to avoid this problem Under this new dependence structure, the main results still hold and extend some related existing ones. Precise large deviations for nonrandom sums and random sums with dominated variation in dependent multi-risk models are presented in Sections 3 and 4

Preliminaries

Large Deviations for Nonrandom Sums

Large Deviations for Random Sums