Approximation of the Tail Probabilities for Bidimensional Randomly Weighted Sums With Dependent Components

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 $\left\{ {{\brTheta} _k = {(\Theta _{1,k},\Theta _{2,k})}^{\top}, k \ge 1} \right\}$ 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 $\left( {\sum\nolimits_{k = 1}^n {\Theta _{1,k}} X_{1,k},\sum\nolimits_{k = 1}^n {\Theta _{2,k}} X_{2,k}} \right)^{\rm \top }$ and their maxima $({{\max} _{1 \le i \le n}}\sum\nolimits_{k = 1}^i {\Theta _{1,k}} X_{1,k},{{\max} _{1 \le i \le n}}\sum\nolimits_{k = 1}^i {\Theta _{2,k}} X_{2,k})^{\rm \top }$ 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.

Exact Results for the Distribution of Randomly Weighted Sums

“…There have been many papers studying the distribution of randomly weighted sums. Papers published since 2020 have studied: inequalities for sums of randomly weighted random variables [1][2][3]; randomly weighted sums of conditionally dependent and dominated varying-tailed increments [4]; second-order tail behavior of randomly weighted heavytailed sums [5]; complete and complete moment convergence for randomly weighted sums [6]; approximations for the tail behavior of bidimensional randomly weighted sums [7,8]; complete convergence for randomly weighted sums of random variables satisfying some moment inequalities [9]; complete convergence and complete moment convergence for maximal randomly weighted sums [10]; asymptotic distributions of randomly weighted sums [11]; complete moment convergence for randomly weighted sums of extended negatively dependent sequences [12]; complete convergence for randomly weighted sums [13,14]; complete f -moment convergence for randomly weighted sums [15]; tail asymptotics of randomly weighted sums of dependent strong subexponential random variables [16]; sums of two dependent randomly weighted random variables [17]; tail behavior of randomly weighted sums of dependent subexponential random variables [18]; randomly weighted sums for multivariate Dirichlet distributions [19]; complete convergence and complete integral convergence for randomly weighted sums [20]; complete moment convergence for randomly weighted sums of negatively superadditive-dependent random variables [21]; complete moment convergence for the maximum of randomly weighted sums [22]; the Baum-Katz theorem for randomly weighted sums [23]; asymptotics for the joint tail probability of bidimensional randomly weighted sums [24]; complete moment convergence for randomly weighted sums [25]. Other highly cited papers studying the distribution of randomly weighted sums include [26][27][28][29][30].…”

Precise large deviations for a multidimensional risk model with regression dependence structure

“…It is well known that, with the increasing diversification of insurance companies' business types, the multidimensional risk model can reflect the influence of different businesses on insurance companies' solvency more comprehensively. Therefore, the risk theory analysis of multidimensional risk model has attracted the attention of some researchers; see, for example, see, Chen et al [5], Loukissas [12], Fu and Liu [8], Lu [13], Shen et al [15], Wang and Wang [19] and references therein.…”

Ruin Probabilities for a Multidimensional Risk Model With Non-Stationary Arrivals and Subexponential Claims

“…, and U m (x) become negative simultaneously and at least one of U i (s) becomes negative, respectively, with inf ∅ = ∞ by convention. Due to the importance in insurance practice, asymptotic estimates for the ruin probability of multidimensional risk models, with various financial and insurance factors, have been widely studied in the past decade; see, for example, Chen et al [5], Chen et al [4], Gao and Yang [8], Konstantinides and Li [11], Li [14], Li et al [16], Li and Yang [15], Liu et al [17], Lu and Zhang [18], Shen et al [19], Yang and Li [21], Yang et al [22], among many others. Notice that, in these literatures, they all assume that {N (t); t ≥ 0} is a renewal counting process, that is, θ i , i ≥ 1, form a sequence of i.i.d.…”