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On the joint tail behavior of randomly weighted sums of heavy-tailed random variables

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  • Li, Jinzhu

Abstract

We focus on the joint tail behavior of randomly weighted sums Sn=U1X1+⋯+UnXn and Tm=V1Y1+⋯+VmYm. The vectors of primary random variables (X1,Y1), (X2,Y2),… are assumed to be independent with dominatedly varying marginal distributions, and the dependence within each pair (Xi,Yi) satisfies a condition called strong asymptotic independence. The random weights U1, V1,… are non-negative and arbitrarily dependent, but they are independent of the primary random variables. Under suitable conditions, we obtain asymptotic expansions for the joint tails of (Sn,Tm) with fixed positive integers n and m, and (SN,TM) with integer-valued random variables N and M that are independent of the primary random variables. When the marginal distributions of the primary random variables are extended regularly varying, the result is proved to hold uniformly for any n and m under stronger conditions. Our results rely critically on moment conditions that are generally easy to check.

Suggested Citation

  • Li, Jinzhu, 2018. "On the joint tail behavior of randomly weighted sums of heavy-tailed random variables," Journal of Multivariate Analysis, Elsevier, vol. 164(C), pages 40-53.
    • Handle: RePEc:eee:jmvana:v:164:y:2018:i:c:p:40-53
    DOI: 10.1016/j.jmva.2017.10.008
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    References listed on IDEAS

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    1. Zhang, Yi & Shen, Xinmei & Weng, Chengguo, 2009. "Approximation of the tail probability of randomly weighted sums and applications," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 655-675, February.
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    Cited by:

    1. Gajek, Lesław & Krajewska, Elżbieta, 2020. "Approximating sums of products of dependent random variables," Statistics & Probability Letters, Elsevier, vol. 164(C).
    2. Richards, Jordan & Tawn, Jonathan A., 2022. "On the tail behaviour of aggregated random variables," Journal of Multivariate Analysis, Elsevier, vol. 192(C).
    3. Chen, Yiqing, 2020. "A Kesten-type bound for sums of randomly weighted subexponential random variables," Statistics & Probability Letters, Elsevier, vol. 158(C).
    4. Lin, Jianxi, 2019. "Second order tail approximation for the maxima of randomly weighted sums with applications to ruin theory and numerical examples," Statistics & Probability Letters, Elsevier, vol. 153(C), pages 37-47.

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