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 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 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 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 Copulas in Classical Probability Sense

2020 ◽  
Vol 17 (3) ◽  
pp. 0889
Author(s):  
Ahmed AL-Adilee ◽  
Zainalabideen Samad ◽  
Samer Al-Shibley
Keyword(s):  
Probability Measure ◽  
Conditional Probability ◽  
Joint Distribution ◽  
Probability Space ◽  
Random Variables ◽  
Distribution Functions ◽  
Classical Probability ◽  
Probability Relation ◽  
Equivalent Probability Measure

               Copulas are simply equivalent structures to joint distribution functions. Then, we propose modified structures that depend on classical probability space and concepts with respect to copulas. Copulas have been presented in equivalent probability measure forms to the classical forms in order to examine any possible modern probabilistic relations. A probability of events was demonstrated as elements of copulas instead of random variables with a knowledge that each probability of an event belongs to [0,1]. Also, some probabilistic constructions have been shown within independent, and conditional probability concepts. A Bay's probability relation and its properties were discussed with respect to copulas. Moreover, an extension of multivariate constructions of each probabilistic copula has been presented. Finally, we have shown some examples that explain each relation of copula in terms of probability space instead of distribution functions.

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.

scholarly journals Explicit results on conditional distributions of generalized exponential mixtures

2020 ◽  
Vol 57 (3) ◽  
pp. 760-774
Author(s):  
Claudia Klüppelberg ◽  
Miriam Isabel Seifert
Keyword(s):  
Risk Management ◽  
Conditional Distribution ◽  
Random Variables ◽  
Threshold Value ◽  
Reliability Theory ◽  
Mixture Distributions ◽  
Tail Behavior ◽  
Conditional Distributions ◽  
Exponential Mixture ◽  
Subset Sum

AbstractFor independent exponentially distributed random variables $X_i$ , $i\in {\mathcal{N}}$ , with distinct rates ${\lambda}_i$ we consider sums $\sum_{i\in\mathcal{A}} X_i$ for $\mathcal{A}\subseteq {\mathcal{N}}$ which follow generalized exponential mixture distributions. We provide novel explicit results on the conditional distribution of the total sum $\sum_{i\in {\mathcal{N}}}X_i$ given that a subset sum $\sum_{j\in \mathcal{A}}X_j$ exceeds a certain threshold value $t>0$ , and vice versa. Moreover, we investigate the characteristic tail behavior of these conditional distributions for $t\to\infty$ . Finally, we illustrate how our probabilistic results can be applied in practice by providing examples from both reliability theory and risk management.

scholarly journals Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 981
Author(s):  
Patricia Ortega-Jiménez ◽  
Miguel A. Sordo ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Financial Risk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Distribution Functions ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Absolute Value ◽  
The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

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