Randomly weighted sums of dependent subexponential random variables with applications to risk theory

For any fixed integer n≥1$ n \ge 1 $, let X1,…,Xn$ X_1,\ldots ,X_n $ be real-valued random variables with a common subexponential distribution, and let θ1,…,θn$ \theta _1,\ldots ,\theta _n $ be positive random variables which are bounded above and independent of X1,…,Xn$ X_1,\ldots ,X_n $. Under some rather loose conditional dependence assumptions on the primary random variables X1,…,Xn$ X_1,\ldots ,X_n $, this paper proves that the asymptotic relationsP∑i=1nθiXi>x∼Pmax1≤m≤n∑i=1mθiXi>x∼Pmax1≤i≤nθiXi>x∼∑i=1nPθiXi>x$$ \begin{aligned} P\left(\sum _{i=1}^n \theta _iX_i >x\right)&\sim P\left( \max _{1\le m\le n}\sum _{i=1}^m \theta _iX_i>x\right)\sim P\left(\max _{1\le i\le n}\theta _iX_i>x\right)\\&\sim \sum _{i=1}^n {P\left( \theta _iX_i>x\right)} \end{aligned} $$hold as x→∞$ x\rightarrow \infty $, where θ1,…,θn$ \theta _1,\ldots ,\theta _n $ are arbitrarily dependent. In particular, it is shown that the above results hold true for X1,…,Xn$ X_1,\ldots ,X_n $ with certain Samarnov distributions. The obtained results on randomly weighted sums are applied to estimating the finite-time ruin probability in a discrete-time risk model with both insurance and financial risks.