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Limit Theorems for Sums of Heavy-tailed Variables with Random Dependent Weights

Author

Listed:
  • Stilian A. Stoev

    (University of Michigan)

  • Murad S. Taqqu

    (Boston University)

Abstract

Let $U_{j} ,\;j \in \mathbb{N}$ be independent and identically distributed random variables with heavy-tailed distributions. Consider a sequence of random weights ${\left\{ {W_{j} } \right\}}_{{j \in \mathbb{N}}}$ , independent of ${\left\{ {U_{j} } \right\}}_{{j \in \mathbb{N}}}$ and focus on the weighted sums ${\sum\nolimits_{j = 1}^{{\left[ {nt} \right]}} {W_{j} {\left( {U_{j} - \mu } \right)}} }$ , where μ involves a suitable centering. We establish sufficient conditions for these weighted sums to converge to non-trivial limit processes, as n→∞, when appropriately normalized. The convergence holds, for example, if ${\left\{ {W_{j} } \right\}}_{{j \in \mathbb{N}}}$ is strictly stationary, dependent, and W 1 has lighter tails than U 1. In particular, the weights W j s can be strongly dependent. The limit processes are scale mixtures of stable Lévy motions. We establish weak convergence in the Skorohod J 1-topology. We also consider multivariate weights and show that they converge weakly in the strong Skorohod M 1-topology. The M 1-topology, while weaker than the J 1-topology, is strong enough for the supremum and infimum functionals to be continuous.

Suggested Citation

  • Stilian A. Stoev & Murad S. Taqqu, 2007. "Limit Theorems for Sums of Heavy-tailed Variables with Random Dependent Weights," Methodology and Computing in Applied Probability, Springer, vol. 9(1), pages 55-87, March.
  • Handle: RePEc:spr:metcap:v:9:y:2007:i:1:d:10.1007_s11009-006-9011-5
    DOI: 10.1007/s11009-006-9011-5
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    References listed on IDEAS

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    1. Ward Whitt, 1980. "Some Useful Functions for Functional Limit Theorems," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 67-85, February.
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    Cited by:

    1. Andriy Olenko & Dareen Omari, 2020. "Reduction Principle for Functionals of Vector Random Fields," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 573-598, June.

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