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 Multiperiod conditional distribution functions for conditionally normal GARCH(1, 1) models

2005 ◽  
Vol 42 (02) ◽  
pp. 426-445
Author(s):  
Raymond Brummelhuis ◽  
Dominique Guégan
Keyword(s):  
Conditional Probability ◽  
Conditional Distribution ◽  
Probability Distributions ◽  
Random Variables ◽  
Distribution Functions ◽  
Tail Behavior ◽  
Asymptotic Tail

We study the asymptotic tail behavior of the conditional probability distributions of r t+k and r t+1+⋯+r t+k when (r t ) t∈ℕ is a GARCH(1, 1) process. As an application, we examine the relation between the extreme lower quantiles of these random variables.

scholarly journals Sums of Dependent Nonnegative Random Variables with Subexponential Tails

2008 ◽  
Vol 45 (1) ◽  
pp. 85-94 ◽  
Author(s):  
Bangwon Ko ◽  
Qihe Tang
Keyword(s):  
Random Variables ◽  
Tail Behavior ◽  
Tail Probabilities ◽  
The Impact ◽  
Asymptotic Tail

In this paper we study the asymptotic tail probabilities of sums of subexponential, nonnegative random variables, which are dependent according to certain general structures with tail independence. The results show that the subexponentiality of the summands eliminates the impact of the dependence on the tail behavior of the sums.

scholarly journals Multiperiod conditional distribution functions for conditionally normal GARCH(1, 1) models

2005 ◽  
Vol 42 (2) ◽  
pp. 426-445 ◽  
Author(s):  
Raymond Brummelhuis ◽  
Dominique Guégan
Keyword(s):  
Conditional Probability ◽  
Conditional Distribution ◽  
Probability Distributions ◽  
Random Variables ◽  
Distribution Functions ◽  
Tail Behavior ◽  
Asymptotic Tail

We study the asymptotic tail behavior of the conditional probability distributions of rt+k and rt+1+⋯+rt+k when (rt)t∈ℕ is a GARCH(1, 1) process. As an application, we examine the relation between the extreme lower quantiles of these random variables.

scholarly journals Sums of Dependent Nonnegative Random Variables with Subexponential Tails

2008 ◽  
Vol 45 (01) ◽  
pp. 85-94 ◽  
Author(s):  
Bangwon Ko ◽  
Qihe Tang
Keyword(s):  
Random Variables ◽  
Tail Behavior ◽  
Tail Probabilities ◽  
The Impact ◽  
Asymptotic Tail

In this paper we study the asymptotic tail probabilities of sums of subexponential, nonnegative random variables, which are dependent according to certain general structures with tail independence. The results show that the subexponentiality of the summands eliminates the impact of the dependence on the tail behavior of the sums.

Tail behavior of the sums of dependent and heavy-tailed random variables

2015 ◽  
Vol 44 (1) ◽  
pp. 12-27 ◽  
Author(s):  
Changjun Yu ◽  
Yuebao Wang ◽  
Dongya Cheng
Keyword(s):  
Random Variables ◽  
Tail Behavior ◽  
Heavy Tailed

scholarly journals Maxima of Sums of Heavy-Tailed Random Variables

2002 ◽  
Vol 32 (1) ◽  
pp. 43-55 ◽  
Author(s):  
K.W. Ng ◽  
Q.H. Tang ◽  
H. Yang
Keyword(s):  
Asymptotic Properties ◽  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Tail Probabilities ◽  
Large Classes ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
The Individual ◽  
Risk Processes

AbstractIn this paper, we investigate asymptotic properties of the tail probabilities of the maxima of partial sums of independent random variables. For some large classes of heavy-tailed distributions, we show that the tail probabilities of the maxima of the partial sums asymptotically equal to the sum of the tail probabilities of the individual random variables. Then we partially extend the result to the case of random sums. Applications to some commonly used risk processes are proposed. All heavy-tailed distributions involved in this paper are supposed on the whole real line.

scholarly journals On the Maximum Exceedance of a Sequence of Random Variables Over a Renewal Threshold

2009 ◽  
Vol 46 (2) ◽  
pp. 559-570 ◽  
Author(s):  
Xuemiao Ha ◽  
Qihe Tang ◽  
Li Wei
Keyword(s):  
Random Walk ◽  
Asymptotic Formula ◽  
Recent Literature ◽  
Random Variables ◽  
Applied Probability ◽  
Tail Behavior ◽  
Heavy Tailed ◽  
Precise Asymptotic ◽  
Independent And Identically Distributed

In this paper we study the tail behavior of the maximum exceedance of a sequence of independent and identically distributed random variables over a random walk. For both light-tailed and heavy-tailed cases, we derive a precise asymptotic formula, which extends and unifies some existing results in the recent literature of applied probability.

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