Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory
Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory
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DOI：
10.1023/b:extr.0000031178.19509.57
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发表时间：
2003-09
期刊：
影响因子：
1.3
通讯作者：
Q. Tang;G. Tsitsiashvili
中科院分区：
文献类型：
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作者：
Q. Tang;G. Tsitsiashvili
设{xk，1≤k≤n}是n个具有公共次指数分布函数的独立实值随机变量，{θk，1≤k≤n}是其他n个独立于{xk，1≤k≤n}且满足a≤θk≤b对某个0<；a≤b<；∞对所有1≤k≤n成立的渐近关系P(max 1≤m≤n∑k=1mθkxk>；x)∼P(≤=1nθkxk>；x)∼sum k=1NP(θkxk>；x)成立。这样，不对序列{θk，1≤k≤n}的依赖结构做出任何假设。给出了它在破产理论中的一个应用。
Let {Xk, 1 ≤k≤n} be n independent and real-valued random variables with common subexponential distribution function, and let {θk, 1 ≤k≤n} be other n random variables independent of {Xk, 1 ≤k≤n} and satisfyinga≤ θk≤bfor some 0 <a≤b< ∞ for all 1 ≤k≤n. This paper proves that the asymptotic relationsP(max1 ≤m≤ n∑k=1mθkXk>x) ∼P(sumk=1nθkXk>x) ∼ sumk=1nP(θkXk>x) hold asx→ ∞. In doing so, no any assumption is made on the dependence structure of the sequence {θk, 1 ≤k≤n}. An application to ruin theory is proposed.