APPROXIMATION OF THE TAIL PROBABILITIES FOR BIDIMENSIONAL RANDOMLY WEIGHTED SUMS WITH DEPENDENT COMPONENTS. Issue 1 (January 2020) - Shen, Xinmei; Ge, Mingyue; Fu, Ke-Ang
Abstract: Let $\left\{ {{\bi X}_k = {(X_{1, k}, X_{2, k})}^{\top}, k \ge 1} \right\}$ be a sequence of independent and identically distributed random vectors whose components are allowed to be generally dependent with marginal distributions being from the class of extended regular variation, and let $\left\{ {{\brTheta} _k = {(Θ _{1, k}, Θ _{2, k})}^{\top}, k \ge 1} \right\}$ be a sequence of nonnegative random vectors that is independent of $\left\{ {{\bi X}_k, k \ge 1} \right\}$ . Under several mild assumptions, some simple asymptotic formulae of the tail probabilities for the bidimensional randomly weighted sums $\left( {\sum\nolimits_{k = 1}^n {Θ _{1, k}} X_{1, k}, \sum\nolimits_{k = 1}^n {Θ _{2, k}} X_{2, k}} \right)^{\rm \top }$ and their maxima $({{\max} _{1 \le i \le n}}\sum\nolimits_{k = 1}^i {Θ _{1, k}} X_{1, k}, {{\max} _{1 \le i \le n}}\sum\nolimits_{k = 1}^i {Θ _{2, k}} X_{2, k})^{\rm \top }$ are established. Moreover, uniformity of the estimate can be achieved under some technical moment conditions on $\left\{ {{\brTheta} _k, k \ge 1} \right\}$ . Direct applications of the results to risk analysis are proposed, with two types of ruin probability for a discrete-time bidimensional risk model being evaluated.