Asymptotics for randomly weighted and stopped dependent sums

References

• Q. Tang, and G. Tsitsiashvili, Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks, Stochastic Process. Appl. 108(2) (2003), pp. 299–325. doi:10.1016/j.spa.2003.07.001.
   Web of Science ®Google Scholar
• Q. Tang, and G. Tsitsiashvili, Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments, Adv. in Appl. Probab. 36(4) (2004), pp. 1278–1299. doi:10.1239/aap/1103662967.
   Web of Science ®Google Scholar
• D. Wang, C. Su, and Y. Zeng, Uniform estimate for maximum of randomly weighted sums with applications to insurance risk theory, Sci. China Ser. A 48 (2005), pp. 1879–1894.
   Google Scholar
• D. Wang, and Q. Tang, Tail probabilities of randomly weighted sums of random variables with dominated variation, Stoch. Models 22(2) (2006), pp. 253–272. doi:10.1080/15326340600649029.
   Web of Science ®Google Scholar
• Y. Zhang, X. Shen, and C. Weng, Approximation of the tail probability of randomly weighted sums and applications, Stochastic Process. Appl. 119(2) (2009), pp. 655–675. doi:10.1016/j.spa.2008.03.004.
   Web of Science ®Google Scholar
• Q. Gao, and Y. Wang, Randomly weighted sums with dominated varying-tailed increments and application to risk theory, J. Korean Statist. Soc. 39(3) (2010), pp. 305–314. doi:10.1016/j.jkss.2010.02.004.
   Web of Science ®Google Scholar
• Y. Chen, and K. Yuen, Sums of pairwise quasi-asymptotically independent random variables with consistent variation, Stoch. Models 25(1) (2009), pp. 76–89. doi:10.1080/15326340802641006.
   Web of Science ®Google Scholar
• L. Yi, Y. Chen, and C. Su, Approximation of the tail probability of randomly weighted sums of dependent random variables with dominated variation, Aust. J. Math. Anal. Appl. 376(1) (2011), pp. 365–372. doi:10.1016/j.jmaa.2010.10.020.
   Web of Science ®Google Scholar
• R.S. Hazra, and K. Maulik, Tail behavior of randomly weighted sums, Adv. in Appl. Probab. 44(3) (2012), pp. 794–814. doi:10.1239/aap/1346955265.
   Web of Science ®Google Scholar
• Y. Yang, R. Leipus, and J. Šiaulys, On the ruin probability in a dependent discrete time risk model with insurance and financial risks, J. Comput. Appl. Math. 236(13) (2012), pp. 3286–3295. doi:10.1016/j.cam.2012.02.030.
   Web of Science ®Google Scholar
• M. Olvera-Cravioto, Asymptotics for weighted random sums, Adv. in Appl. Probab. 44(4) (2012), pp. 1142–1172. doi:10.1239/aap/1354716592.
   Web of Science ®Google Scholar
• Y. Volkovich, and N. Litvak, Asymptotic analysis for personalized web search, Adv. in Appl. Probab. 42(2) (2010), pp. 577–604. doi:10.1239/aap/1275055243.
   Web of Science ®Google Scholar
• P.R. Jelenković, and M. Olvera-Cravioto, Information ranking and power laws on trees, Adv. Appl. Probab. 42(4) (2010), pp. 1057–1093. doi:10.1239/aap/1293113151.
   Web of Science ®Google Scholar
• N.H. Bingham, C.M. Goldie, and J.L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.
   Google Scholar
• L. Liu, Precise large deviations for dependent random variables with heavy tails, Stat. Probab. Lett. 79(9) (2009), pp. 1290–1298. doi:10.1016/j.spl.2009.02.001.
   Web of Science ®Google Scholar
• N. Ebrahimi, and M. Ghosh, Multivariate negative dependence, Comm. Statist. Theory Methods 10(4) (1981), pp. 307–337. doi:10.1080/03610928108828041.
   Web of Science ®Google Scholar
• H.W. Block, T.H. Savits, and M. Shaked, Some concepts of negative dependence, Ann. Probab. 10(3) (1982), pp. 765–772. doi:10.1214/aop/1176993784.
   Web of Science ®Google Scholar
• K. Wang, Y. Wang, and Q. Gao, Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate, Methodol. Comput. Appl. Probab. 15(1) (2013), pp. 109–124. doi:10.1007/s11009-011-9226-y.
   Web of Science ®Google Scholar
• Y. Chen, A. Chen, and K.W Ng, The strong law of large numbers for extended negatively dependent random variables, J. Appl. Probab. 47(4) (2010), pp. 908–922. doi:10.1239/jap/1294170508.
   Web of Science ®Google Scholar
• M. Sibuya, Bivariate extreme statistics, I, Ann. Inst. Statist. Math. 11(2) (1959), pp. 195–210. doi:10.1007/BF01682329.
   Google Scholar
• M. De Carvalho, and A. Ramos, Bivariate extreme statistics, II, REVSTAT 10 (2012), pp. 83–107.
   Web of Science ®Google Scholar
• X. Liu, Q. Gao, and Y. Wang, A note on a dependent risk model with constant interest rate, Stat. Probab. Lett. 82(4) (2012), pp. 707–712. doi:10.1016/j.spl.2011.12.016.
   Web of Science ®Google Scholar
• B. Ko, and Q. Tang, Sums of dependent nonnegative random variables with subexponential tails, J. Appl. Probab. 45(1) (2008), pp. 85–94. doi:10.1239/jap/1208358953.
   Web of Science ®Google Scholar
• L. Dindienė, and R Leipus, A note on the tail behavior of randomly weighted and stopped dependent sums, Nonlinear Anal. Model. Control 20 (2015), pp. 263–273.
   Web of Science ®Google Scholar
• D. B. H Cline, and G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stochastic Process. Appl. 49(1) (1994), pp. 75–98. doi:10.1016/0304-4149(94)90113-9.
   Web of Science ®Google Scholar
• Q. Tang, and G. Tsitsiashvili, Randomly weighted sums of subexponential random variables with application to ruin theory, Extremes 6(3) (2003), pp. 171–188. doi:10.1023/B:EXTR.0000031178.19509.57.
   Google Scholar
• M. Zhou, K. Wang, and Y. Wang, Estimates for the finite-time ruin probability with insurance and financial risks, Acta Math. Appl. Sin. Engl. Ser. 28(4) (2012), pp. 795–806. doi:10.1007/s10255-012-0189-8.
   Web of Science ®Google Scholar
• Y. Yang, and Y. Wang, Tail behavior of the product of two dependent random variables with applications to risk theory, Extremes 16(1) (2013), pp. 55–74. doi:10.1007/s10687-012-0153-2.
   Web of Science ®Google Scholar
• Q. Tang, Asymptotics for the finite time ruin probability in the renewal model with consistent variation, Stoch.Models 20(3) (2004), pp. 281–297. doi:10.1081/STM-200025739.
   Web of Science ®Google Scholar
• Y. Wang, Q. Gao, K. Wang, and X. Liu, Random time ruin probability for the renewal risk model with heavy-tailed claims, J. Ind. Manag. Optim. 5(4) (2009), pp. 719–736. doi:10.3934/jimo.2009.5.719.
   Web of Science ®Google Scholar
• Y. Wang, Z. Cui, K. Wang, and X. Ma, Uniform asymptotics of the finite-time ruin probability for all times, J. Math. Anal. Appl. 390(1) (2012), pp. 208–223. doi:10.1016/j.jmaa.2012.01.025.
   Web of Science ®Google Scholar
• X. Bai, and L. Song, Asymptotic behavior of random time ruin probability under heavy-tailed claim sizes and dependence structure, Comm. Statist. Theory Methods 41(10) (2012), pp. 1721–1732. doi:10.1080/03610926.2010.549992.
   Web of Science ®Google Scholar
• X. Bai, and L. Song, Asymptotic behavior of random time absolute ruin probability with tailed and conditionally independent claim sizes, Stat. Probab. Lett. 82(9) (2012), pp. 1718–1726. doi:10.1016/j.spl.2012.05.010.
   Web of Science ®Google Scholar