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Asymptotics of randomly weighted sums without moment conditions of random weights

Qingwu Gao, Dimitrios G. Konstantinides, Charalampos D. Passalidis, Yuebao Wang, Hui Xu

Abstract

In the paper, under a suitable condition, we study the asymptotics of randomly weighted sums and randomly weighted stopped sums with upper tail asymptotically independent increments, where no moment condition is made on random weights, and we provide an example to show the suitable condition is necessary in some sense. To this end, we firstly consider the uniform asymptotics of the corresponding weighted sums with a large convergence range than the existing results. Then, using the above results, we obtain an asymptotic estimation of the finite-time and random-time ruin probabilities in a discrete-time risk model. In the case of regular variation increments, a more explicit estimation is given by an extended Breiman's theorem. Finally, through some examples we illustrate that the conditions of the above results are relaxed and clear, and that there exist random variables are upper tail asymptotically independent rather than tail asymptotically independent.

Asymptotics of randomly weighted sums without moment conditions of random weights

Abstract

In the paper, under a suitable condition, we study the asymptotics of randomly weighted sums and randomly weighted stopped sums with upper tail asymptotically independent increments, where no moment condition is made on random weights, and we provide an example to show the suitable condition is necessary in some sense. To this end, we firstly consider the uniform asymptotics of the corresponding weighted sums with a large convergence range than the existing results. Then, using the above results, we obtain an asymptotic estimation of the finite-time and random-time ruin probabilities in a discrete-time risk model. In the case of regular variation increments, a more explicit estimation is given by an extended Breiman's theorem. Finally, through some examples we illustrate that the conditions of the above results are relaxed and clear, and that there exist random variables are upper tail asymptotically independent rather than tail asymptotically independent.
Paper Structure (15 sections, 13 theorems, 137 equations)

This paper contains 15 sections, 13 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Dependence structures
  3. Distribution classes
  4. Main results
  5. Proofs of Theorem \ref{['thm101']} and Theorem \ref{['thm102']}
  6. Proofs of Theorem \ref{['thm101']}
  7. Proofs of Theorem \ref{['thm102']}
  8. Applications to risk theory
  9. Finite-time ruin probability
  10. Random-time ruin probability
  11. Some examples
  12. TAI structure and UTAI structure
  13. On condition (\ref{['10800']})
  14. On functions $f_1(\cdot)$ and $f_2(\cdot)$
  15. On the long-tail property of $I(\cdot)$

Key Result

Proposition 1.1

(1) R.v.s $\xi_i,i\ge1,$ are TAI $\Rightarrow$ they are UTAI, but otherwise. (2) If $\xi_i,i\ge1,$ are nonnegative r.v.s, then they are TAI $\Leftrightarrow$ they are UTAI. (3) R.v.s $\xi_i,i\ge1,$ are UTAI $\Leftrightarrow$ r.v.s $\xi^+_i,i\ge1,$ are UTAI.

Theorems & Definitions (23)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Corollary 1.1
  • Theorem 3.1
  • ...and 13 more