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Exact Results for the Distribution of Randomly Weighted Sums

Author

Listed:
  • Thomas Hitchen

    (Department of Mathematics, University of Manchester, Manchester M13 9PL, UK)

  • Saralees Nadarajah

    (Department of Mathematics, University of Manchester, Manchester M13 9PL, UK)

Abstract

Dependent random variables play a crucial role in various fields, from finance and statistics to engineering and environmental sciences. Often, interest lies in understanding the aggregate sum of a collection of dependent variables with random weights. In this paper, we provide a comprehensive study of the distribution of the aggregate sum with random weights. Expressions derived include those for the cumulative distribution function, probability density function, conditional expectation, moment generating function, characteristic function, cumulant generating function, moments, variance, skewness, kurtosis, cumulants, value at risk and the expected shortfall. Real data applications are discussed.

Suggested Citation

  • Thomas Hitchen & Saralees Nadarajah, 2024. "Exact Results for the Distribution of Randomly Weighted Sums," Mathematics, MDPI, vol. 12(1), pages 1-22, January.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:1:p:149-:d:1312098
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    References listed on IDEAS

    as
    1. Nan Cheng & Chao Lu & Jibing Qi & Xuejun Wang, 2022. "Complete moment convergence for randomly weighted sums of extended negatively dependent random variables with application to semiparametric regression models," Statistical Papers, Springer, vol. 63(2), pages 397-419, April.
    2. 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.
    3. Xinmei Shen & Kailin Du, 2023. "Uniform Approximation for the Tail Behavior of Bidimensional Randomly Weighted Sums," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-25, March.
    4. Jianxi Lin, 2020. "Second order tail behaviour of randomly weighted heavy-tailed sums and their maxima," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 49(11), pages 2648-2670, June.
    5. Dawei Lu & Jialu Wang, 2021. "Complete convergence and complete moment convergence for maximal randomly weighted sums of widely orthant-dependent random variables with applications," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 50(4), pages 763-791, February.
    6. Chen, Yiqing, 2020. "A Kesten-type bound for sums of randomly weighted subexponential random variables," Statistics & Probability Letters, Elsevier, vol. 158(C).
    7. Xiu Xu & Jigao Yan, 2021. "Complete moment convergence for randomly weighted sums of END sequences and its applications," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 50(12), pages 2877-2899, June.
    8. Roozegar, Rasool & zarch, Hamid Reza Taherizadeh, 2021. "On the asymptotic distribution of randomly weighted averages of random vectors," Statistics & Probability Letters, Elsevier, vol. 179(C).
    Full references (including those not matched with items on IDEAS)

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