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

Published online by Cambridge University Press:  05 December 2018

Xinmei Shen ,
Mingyue Ge  and
Ke-Ang Fu
Xinmei Shen
Affiliation:
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China E-mail: xshen@dlut.edu.cn; gemingyue@mail.dlut.edu.cn
Mingyue Ge
Affiliation:
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China E-mail: xshen@dlut.edu.cn; gemingyue@mail.dlut.edu.cn
Ke-Ang Fu
Affiliation:
School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China E-mail: fukeang@hotmail.com
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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.

Keywords

asymptoticsbidimensional randomly weighted sumscopulaextended regular variation
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Issue 1 , January 2020 , pp. 112 - 130
Copyright
Copyright © Cambridge University Press 2018 

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