Second-order Asymptotic Analysis of Tail Probabilities of Bidimensional Randomly Weighted Sums

Abstract

Motivated by a bidimensional discrete-time risk model in insurance, we study 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 $S_n^{\xi}=\sum_{i=1}^n\xi_iX_i$ and $T_m^\eta=\sum_{j=1}^m\eta_{j}Y_{j}$ for any fixed $n,m\in\N$, in which it is assumed that the primary random variables $\left\{\(X,Y\),\(X_i,Y_i\):i\in \N\right\}$ form a sequence of real-valued, independent and identically distributed random pairs following a common bivariate Farlie-Gumbel-Morgenstern distribution and the random weights $\left\{\xi_i,\eta_i:i\in \N\right\}$ 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 asymptotic formulas for the joint and sum tail probabilities, which generalize and strengthen some known ones in the literature. Furthermore, by directly applying the obtained results to the above bidimensional risk model, we establish second-order asymptotic formulas for the corresponding tail probabilities. Compared with the first-order ones, our numerical simulation shows that the second-order asymptotics are much more precise.