APPROXIMATION OF THE TAIL PROBABILITIES FOR BIDIMENSIONAL RANDOMLY WEIGHTED SUMS WITH DEPENDENT COMPONENTS

Let {X-k = (X-1,X-k, X-2,X-k)(inverted perpendicular), 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 {Theta(k) = (Theta(1,k), Theta(2,k))(inverted perpendicular), 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 (Sigma(n)(k=1) Theta X-1,k(1,k), Sigma(n)(k=1) Theta X-2,k(2,k))(inverted perpendicular) and their maxima (max(1 <= i <= n) Sigma(i)(k=1) Theta X-1,k(1,k), max(1 <= i <= n) Sigma(i)(k=1) Theta X-2,k(2,k))(inverted perpendicular) are established. Moreover, uniformity of the estimate can be achieved under some technical moment conditions on {Theta(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.