Second-order Asymptotic Analysis of Tail Probabilities of Randomly Weighted Sums: With Applications to a Bidimensional Discrete-time Risk Model
Bingzhen Geng, Yang Liu, Shijie Wang
TL;DR
This work addresses precise tail risk in a two-line insurance setting by studying second-order asymptotics for randomly weighted sums. It assumes i.i.d. claim pairs $(X,Y)$ with marginals $F,G$ in the second-order subexponential class and a bidimensional FGM dependence with parameter $r\in[-1,1]$, while the stochastic weights $(\theta_i,\Theta_i)$ are bounded, nonnegative, and independent of the claims. The authors derive second-order expansions for the joint tail probability $P(S_n^\theta>x, T_m^\Theta>y)$ and the sum tail probability $P(S_n^\theta+T_m^\Theta>x+y)$, detailing the first-order terms, multiple second-order corrections, and uniform remainder bounds; they also provide corollaries under density assumptions that yield explicit expressions. An actuarial application to a bidimensional discrete-time risk model is included, together with numerical evidence showing that second-order asymptotics substantially improve accuracy over first-order results. The results enhance risk assessment for multi-line insurance portfolios and offer a foundation for extending second-order tail analysis to more complex multi-dimensional risk models.
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.