### Abstract

We are interested in the tail behavior of the randomly weighted sum ∑_{i=1} ^{n} θ_{i} X_{i}, in which the primary random variables X_{1}, …, X_{n} are real valued, independent and subexponentially distributed, while the random weights θ_{1}, …, θ_{n} are nonnegative and arbitrarily dependent, but independent of X_{1}, …, X_{n}. For various important cases, we prove that the tail probability of ∑_{i=1} ^{n} θ_{i}X_{i} is asymptotically equivalent to the sum of the tail probabilities of θ_{1}X_{1}, …, θ_{n}X_{n}, which complies with the principle of a single big jump. An application to capital allocation is proposed.

Original language | English (US) |
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Pages (from-to) | 467-493 |

Number of pages | 27 |

Journal | Extremes |

Volume | 17 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 2014 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Engineering (miscellaneous)
- Economics, Econometrics and Finance (miscellaneous)

### Cite this

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*Extremes*, vol. 17, no. 3, pp. 467-493. https://doi.org/10.1007/s10687-014-0191-z

**Randomly weighted sums of subexponential random variables with application to capital allocation.** / Tang, Qihe; Yuan, Zhongyi.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Randomly weighted sums of subexponential random variables with application to capital allocation

AU - Tang, Qihe

AU - Yuan, Zhongyi

PY - 2014/9/1

Y1 - 2014/9/1

N2 - We are interested in the tail behavior of the randomly weighted sum ∑i=1 n θi Xi, in which the primary random variables X1, …, Xn are real valued, independent and subexponentially distributed, while the random weights θ1, …, θn are nonnegative and arbitrarily dependent, but independent of X1, …, Xn. For various important cases, we prove that the tail probability of ∑i=1 n θiXi is asymptotically equivalent to the sum of the tail probabilities of θ1X1, …, θnXn, which complies with the principle of a single big jump. An application to capital allocation is proposed.

AB - We are interested in the tail behavior of the randomly weighted sum ∑i=1 n θi Xi, in which the primary random variables X1, …, Xn are real valued, independent and subexponentially distributed, while the random weights θ1, …, θn are nonnegative and arbitrarily dependent, but independent of X1, …, Xn. For various important cases, we prove that the tail probability of ∑i=1 n θiXi is asymptotically equivalent to the sum of the tail probabilities of θ1X1, …, θnXn, which complies with the principle of a single big jump. An application to capital allocation is proposed.

UR - http://www.scopus.com/inward/record.url?scp=84957428639&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84957428639&partnerID=8YFLogxK

U2 - 10.1007/s10687-014-0191-z

DO - 10.1007/s10687-014-0191-z

M3 - Article

AN - SCOPUS:84957428639

VL - 17

SP - 467

EP - 493

JO - Extremes

JF - Extremes

SN - 1386-1999

IS - 3

ER -