Asymptotics for the joint tail probability of bidimensional randomly weighted sums with applications to insurance

This paper studies the joint tail behavior of two randomly weighted sums∑_(i=1)^(m)Θ_(i)X_(i)and∑_(j=1)^(n)θ_(j)Y_(j)for some m,n∈N∪{∞},in which the primary random variables{X_(i);i∈N}and{Y_(i);i∈N},respectively,are real-valued,dependent and heavy-tailed,while the random weights{Θi,θi;i∈N}are nonnegative and arbitrarily dependent,but the three sequences{X_(i);i∈N},{Y_(i);i∈N}and{Θ_(i),θ_(i);i∈N}are mutually independent.Under two types of weak dependence assumptions on the heavy-tailed primary random variables and some mild moment conditions on the random weights,we establish some(uniformly)asymptotic formulas for the joint tail probability of the two randomly weighted sums,expressing the insensitivity with respect to the underlying weak dependence structures.As applications,we consider both discrete-time and continuous-time insurance risk models,and obtain some asymptotic results for ruin probabilities.

Uniform Estimate for The Tail Probabilities of Randomly Weighted Sums

Several authors have studied the uniform estimate for the tail probabilities of randomly weighted sumsa.ud their maxima. In this paper, we generalize their work to the situation thatis a sequence of upper tail asymptotically independent random variables with common distribution from the is a sequence of nonnegative random variables, independent of and satisfying some regular conditions. Moreover. no additional assumption is required on the dependence structureof {θi,i≥ 1）.

A Note on Randomly Weighted Sums of Dependent Subexponential Random Variables

The author obtains that the asymptotic relations ■ hold as x→∞, where the random weights θ1,···, θn are bounded away both from 0 and from∞with no dependency assumptions, independent of the primary random variables X1,···, Xn which have a certain kind of dependence structure and follow non-identically subexponential distributions. In particular, the asymptotic relations remain true when X1,···, Xn jointly follow a pairwise Sarmanov distribution.

Asymptotic Tail Probability of Randomly Weighted Sums of Dependent Random Variables with Dominated Variation

This paper investigates the asymptotic behavior of tail probability of randomly weighted sums of dependent and real-valued random variables with dominated variation, where the weights form another sequence of nonnegative random variables. The result we obtain extends the corresponding result of Wang and Tang.

Complete Convergence for Randomly Weighted Sums of Random Variables Satisfying Some Moment Inequalities

For random variables and random weights satisfying Marcinkiewicz-Zygmund and Rosenthal type moment inequalities, we establish complete convergence results for randomly weighted sums of the random variables. Our results generalize those of(Thanh et al. SIAM J. Control Optim., 49,106–124(2011), Han and Xiang J. Ineq. Appl., 2016, 313(2016), Li et al. J. Ineq. Appl., 2017, 182(2017), and Wang et al. Statistics, 52, 503–518(2018).)