Approximation Of The Tail Probabilities For Bidimensional Randomly Weighted Sums And Its Application

In this paper,approximation of the tail probabilities for bidimensional randomly weighted sums are discussed.The main contents include the following aspects.Firstly,some definitions of the classes of the heavy-tailed distribution,the definition of Copula and the current status of related research are briefly introduced.Secondly,approximation of the tail probabilities for bidimensional randomly weight-ed sums is focused.Let {Xk=(Xl.k,X2.k)T,k≥ 1} be a sequence of independent and i-dentically distributed random vectors whose components are allowed to be generally depen-dent with marginal distributions being from the class of extended regular variation,and let{(?)k=((?)1,k,(?)2.k)T,k≥1} be a sequence of nonnegative random vectors that is independent of {Xk,k>1}.Under several mild assumptions,some simple asymptotic formula of the tail probabilities for the bidimensional randomly weighted sums(∑nk=1(?)1,kX1,k,∑nk=1(?)2,kX2,K)T and their maxima(max1≤i≤n=∑ik=1(?)1,k,max1≤i≤n ∑ik=1(?)2,k X2,k X2,k)T are established.Thirdly,under the previous chapter,uniformity of the estimate can be achieved under some technical moment conditions on {(?)k,k>1}.Fourthly,we consider a stochastic economic environment,based on Chapter 2 and Chapter 3,direct applications of the results to risk analysis are proposed,with two types of ruin proba-bility for a discrete-time bidimensional risk model being evaluated.Finally,under some technical moment conditions,random variables N is independent with{(?)k,k≥1} and {Xk,k≥ 1},the asymptotic estimation of the tail probability for weighted randomly sums is obtained.