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 { X_k = (X_1, k, X_2, k)^, k 1 } 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 { _k = (Θ _1, k, Θ _2, k)^, k 1 } be a sequence of nonnegative random vectors that is independent of { X_k, k 1 } . Under several mild assumptions, some simple asymptotic formulae of the tail probabilities for the bidimensional randomly weighted sums ( olimits_k = 1^n Θ _1, k X_1, k, olimits_k = 1^n Θ _2, k X_2, k )^ and their maxima ( _1 i nolimits_k = 1^i Θ _1, k X_1, k, _1 i nolimits_k = 1^i Θ _2, k X_2, k)^ are established. Moreover, uniformity of the estimate can be achieved under some technical moment conditions on { _k, k 1 } . 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.