Second-order Asymptotic Analysis of Tail Probabilities of Randomly Weighted Sums: With Applications to a Bidimensional Discrete-time Risk Model

TL;DR

This paper derives second-order asymptotic formulas for tail probabilities of randomly weighted sums in a bidimensional risk model, enhancing the accuracy of risk assessment in insurance applications.

Contribution

It provides the first second-order asymptotic analysis for tail probabilities of randomly weighted sums with dependent weights and bivariate distributions, extending existing first-order results.

Findings

Second-order asymptotics significantly improve approximation accuracy.

Results apply to bidimensional risk models with dependent weights.

Numerical simulations confirm the enhanced precision of second-order formulas.

Abstract

Motivated by a bidimensional discrete-time risk model in insurance, we study the second-order asymptotics for two kinds of tail probabilities of the stochastic discounted value of aggregate net losses including two business lines. These are essentially modeled as randomly weighted sums, in which it is assumed that the primary random variables form a sequence of real-valued, independent and identically distributed random pairs following a common bivariate Farlie-Gumbel-Morgenstern distribution and the random weights are bounded, nonnegative and arbitrarily dependent, but independent of the primary random variables. Under the assumption that two marginal distributions of the primary random variables are second-order subexponential, we first obtain the second-order asymptotics for the joint and sum tail probabilities, which generalizes and strengthens some known ones in the literature. Furthermore, by directly applying the obtained results to the above bidimensional risk model, we establish the second-order asymptotic formulas for the corresponding tail probabilities. Compared with the first-order one, our numerical simulation shows that second-order asymptotics are much more precise.