Asymptotic Tail Probability of Randomly Weighted Sum of Dependent Heavy-Tailed Random Variables

2010 ◽  
Vol 4 (2) ◽  
Author(s):  
Yu Chen ◽  
Weiping Zhang ◽  
Jie Liu
Keyword(s):  
Random Variables ◽  
Tail Probability ◽  
Weighted Sum ◽  
Heavy Tailed ◽  
Asymptotic Tail Probability ◽  
Asymptotic Tail

A note on the max-sum equivalence of randomly weighted sums of heavy-tailed random variables

2013 ◽  
Vol 18 (4) ◽  
pp. 519-525 ◽  
Author(s):  
Yang Yang ◽  
Kaiyong Wang ◽  
Remigijus Leipus ◽  
Jonas Šiaulys
Keyword(s):  
Random Variables ◽  
Tail Probability ◽  
Weighted Sums ◽  
Asymptotic Formulas ◽  
Dependence Structure ◽  
Insurance Risk ◽  
Risk Models ◽  
Randomly Weighted Sums ◽  
Random Weights ◽  
Heavy Tailed

This paper investigates the asymptotic behavior for the tail probability of the randomly weighted sums ∑k=1nθkXk and their maximum, where the random variables Xk and the random weights θk follow a certain dependence structure proposed by Asimit and Badescu [1] and Li et al. [2]. The obtained results can be used to obtain asymptotic formulas for ruin probability in the insurance risk models with discounted factors.

scholarly journals Tail behavior of negatively associated heavy-tailed sums

2006 ◽  
Vol 43 (02) ◽  
pp. 587-593 ◽  
Author(s):  
Jaap Geluk ◽  
Kai W. Ng
Keyword(s):  
Random Variables ◽  
Distribution Functions ◽  
Subexponential Distribution ◽  
Tail Behavior ◽  
Tail Probabilities ◽  
Negatively Associated ◽  
Class D ◽  
Dominatedly Varying Tails ◽  
Heavy Tailed ◽  
Asymptotic Tail

Consider a sequence {X k , k ≥ 1} of random variables on (−∞, ∞). Results on the asymptotic tail probabilities of the quantities , and S (n) = max0 ≤ k ≤ n S k , with X 0 = 0 and n ≥ 1, are well known in the case where the random variables are independent with a heavy-tailed (subexponential) distribution. In this paper we investigate the validity of these results under more general assumptions. We consider extensions under the assumptions of having long-tailed distributions (the class L) and having the class D ∩ L, where D is the class of distribution functions with dominatedly varying tails. Some results are also given in the case where X k , k ≥ 1, are not necessarily identically distributed and/or independent.

scholarly journals Tail behavior of negatively associated heavy-tailed sums

2006 ◽  
Vol 43 (2) ◽  
pp. 587-593 ◽  
Author(s):  
Jaap Geluk ◽  
Kai W. Ng
Keyword(s):  
Random Variables ◽  
Distribution Functions ◽  
Subexponential Distribution ◽  
Tail Behavior ◽  
Tail Probabilities ◽  
Negatively Associated ◽  
Class D ◽  
Dominatedly Varying Tails ◽  
Heavy Tailed ◽  
Asymptotic Tail

Consider a sequence {Xk, k ≥ 1} of random variables on (−∞, ∞). Results on the asymptotic tail probabilities of the quantities , and S(n) = max0 ≤ k ≤ nSk, with X0 = 0 and n ≥ 1, are well known in the case where the random variables are independent with a heavy-tailed (subexponential) distribution. In this paper we investigate the validity of these results under more general assumptions. We consider extensions under the assumptions of having long-tailed distributions (the class L) and having the class D ∩ L, where D is the class of distribution functions with dominatedly varying tails. Some results are also given in the case where Xk, k ≥ 1, are not necessarily identically distributed and/or independent.

scholarly journals Improving the Asmussen–Kroese-Type Simulation Estimators

2012 ◽  
Vol 49 (4) ◽  
pp. 1188-1193 ◽  
Author(s):  
Samim Ghamami ◽  
Sheldon M. Ross
Keyword(s):  
Monte Carlo ◽  
Nonnegative Integer ◽  
Rare Event ◽  
Random Variables ◽  
Random Variable ◽  
Heavy Tailed ◽  
Independent Identically Distributed

The Asmussen–Kroese Monte Carlo estimators of P(Sn > u) and P(SN > u) are known to work well in rare event settings, where SN is the sum of independent, identically distributed heavy-tailed random variables X1,…,XN and N is a nonnegative, integer-valued random variable independent of the Xi. In this paper we show how to improve the Asmussen–Kroese estimators of both probabilities when the Xi are nonnegative. We also apply our ideas to estimate the quantity E[(SN-u)+].

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