Hostname: page-component-78857b5c4d-nbnlr Total loading time: 0 Render date: 2023-08-04T00:24:39.269Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": false, "useRatesEcommerce": true } hasContentIssue false

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

Published online by Cambridge University Press:  05 December 2018

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

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.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bingham, N.H., Goldie, C.M. & Teugels, J.L. (1987). Regular Variation Cambridge: Cambridge University Press.CrossRefGoogle Scholar
2.Chen, Y.Q., Ng, K.W. & Xie, X.S. (2006). On the maximum of randomly weighted sums with regularly varying tails. Statistics and Probability Letters 76: 971975.CrossRefGoogle Scholar
3.Chen, Y.Q., Yuen, K.C. & Ng, K.W. (2011). Asymptotics for the ruin probabilities of a two-dimensional renewal risk model with heavy-tailed claims. Applied Stochastic Models in Business and Industry 27: 290300.CrossRefGoogle Scholar
4.Chen, Y., Wang, Y. & Wang, K. (2013). Asymptotic results for ruin probability of a two-dimensional renewal risk model. Stochastic Analysis and Applications 31(1): 8091.CrossRefGoogle Scholar
5.Cline, D.B.H. & Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stochastic Processes and their Applications 49(1): 7598.CrossRefGoogle Scholar
6.Gao, Q. & Wang, Y. (2010). Randomly weighted sums with dominated varying-tailed increments and application to risk theory. The Journal of the Korean Statistical Society 39: 305314.CrossRefGoogle Scholar
7.Hazra, R.S. & Maulik, K. (2012). Tail behavior of randomly weighted sums. Advances in Applied Probability 44: 794814.CrossRefGoogle Scholar
8.Hu, Z. & Jiang, B. (2013). On joint ruin probabilities of a two-dimensional risk model with constant interest rate. Journal of Applied Probability 50(2): 309322.CrossRefGoogle Scholar
9.Huang, W., Weng, C.G. & Zhang, Y. (2014). Multivariate risk models under heavy-tailed risks. Applied Stochastic Models in Business and Industry 30: 341360.CrossRefGoogle Scholar
10.Li, J. (2016). Uniform asymptotics for a multi-dimensional time-dependent risk model with multivariate regularly varying claims and stochastic return. Insurance: Mathematics and Economics 71: 195204.Google Scholar
11.Li, J. (2017). The infinite-time ruin probability for a bidimensional renewal risk model with constant force of interest and dependent claims. Communications in Statistics - Theory and Methods 46: 19591971.CrossRefGoogle Scholar
12.Li, J. (2018). On the joint tail behavior of randomly weighted sums of heavy-tailed random variables. Journal of Multivariate Analysis 164: 4053.CrossRefGoogle Scholar
13.Li, J., Liu, Z. & Tang, Q. (2007). On the ruin probabilities of a bidimensional perturbed risk model. Insurance: Mathematics and Economics 41: 185195.Google Scholar
14.Olvera-Cravioto, M. (2012). Asymptotics for weighted random sums. Advances in Applied Probability 44: 11421172.CrossRefGoogle Scholar
15.Shen, X.M. & Zhang, Y. (2013). Ruin probabilities of a two-dimensional risk model with dependent risks of heavy tail. Statistics and Probability Letters 83: 17871799.CrossRefGoogle Scholar
16.Shen, X.M., Lin, Z.Y. & Zhang, Y. (2009). Uniform estimate for maximum of randomly weighted sums with applications to ruin theory. Methodology and Computing in Applied Probability 11: 669685.CrossRefGoogle Scholar
17.Tang, Q. & Tsitsiashvili, G. (2003). Randomly weighted sums of subexponential random variables with application to ruin theory. Extremes 6: 171188.CrossRefGoogle Scholar
18.Tang, Q. & Tsitsiashvili, G. (2004). Finite and infinite time ruin probabilities in the presence of stochastic returns on investments. Advances in Applied Probability 36(4): 12781299.CrossRefGoogle Scholar
19.Tang, Q. & Yuan, Z. (2014). Randomly weighted sums of subexponential random variables with application to capital allocation. Extremes 17: 467493.CrossRefGoogle Scholar
20.Yang, H. & Li, J. (2014). Asymptotic finite-time ruin probability for a bidimensional renewal risk model with constant interest force and dependent subexponential claims. Insurance: Mathematics and Economics 58: 185192.Google Scholar
21.Zhang, Y. & Wang, W. (2012). Ruin probabilities of a bidimensional risk model with investment. Statistics and Probability Letters 82(1): 130138.CrossRefGoogle Scholar
22.Zhang, Y., Shen, X.M. & Weng, C.G. (2009). Approximation of the tail probability of randomly weighted sums and applications. Stochastic Processes and their Applications 119: 655675.CrossRefGoogle Scholar
23.Zhou, M., Wang, K. & Wang, Y. (2012). Estimates for the finite time ruin probability with insurance and financial risks. Acta Mathematicae Applicatae Sinica (English Series) 28: 795806.CrossRefGoogle Scholar