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ABSTRACT: This paper obtains some equivalent conditions about the asymptotics for the density of the supremum of a random walk with
light-tailed increments in the intermediate case. To do this, the paper first corrects the proofs of some existing results
about densities of random sums. On the basis of the above results, the paper obtains some equivalent conditions about the
asymptotics for densities of ruin distributions in the intermediate case and densities of infinitely divisible distributions.
In the above studies, some differences and relations between the results on a distribution and its corresponding density can
be discovered.
Journal of Theoretical Probability 05/2012; 22(2):281-293. · 0.68 Impact Factor
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ABSTRACT: This paper gives an asymptotically equivalent formula for the finite-time ruin probability of a nonstandard risk model with
a constant interest rate, in which both claim sizes and inter-arrival times follow a certain dependence structure. This new
dependence structure allows the underlying random variables to be either positively or negatively dependent. The obtained
asymptotics hold uniformly in a finite time interval. Especially, in the renewal risk model the uniform asymptotics of the
finite-time ruin probability for all times have been given. The obtained results have extended and improved some corresponding
results.
KeywordsUniform asymptotics–Finite-time ruin probability–Constant interest rate–Widely orthant dependent
Methodology And Computing In Applied Probability 04/2012; · 0.75 Impact Factor
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ABSTRACT: In this paper, we consider dependent random variables X
k
, k=1,2,… with supports on [−b
k
,∞), respectively, where the b
k
≥0 are some finite constants. We derive asymptotic results on the tail probabilities of the quantities Sn=åk=1n XkS_{n}=\sum_{k=1}^{n} X_{k}, X
(n)=max 1≤k≤n
X
k
and S
(n)=max 1≤k≤n
S
k
, n≥1 in the case where the random variables are dependent with heavy-tailed (subexponential) distributions, which substantially
generalize the results of Ko and Tang (J. Appl. Probab. 45, 85–94, 2008).
KeywordsAsymptotic tail probability–Convolution–Dependence–Subexponentiality
Acta Applicandae Mathematicae 04/2012; 114(3):219-231. · 0.90 Impact Factor
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ABSTRACT: In 2007, Chen and Ng investigated infinite-time ruin probability with constant interest force and negatively quadrant dependent
and extended regularly varying-tailed claims. Following this work, the authors obtain a weakly asymptotic equivalent formula
for the finite-time and infinite-time ruin probability with constant interest force, negatively quadrant dependent, and dominated
varying-tailed claims and negatively lower orthant dependent inter-arrival times. In particular, when the claims are consistently
varying-tailed, an asymptotic equivalent formula is presented.
Journal of Systems Science and Complexity 04/2012; 22(3):407-414. · 0.37 Impact Factor
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ABSTRACT: Under some relaxed conditions, in this paper we obtain some equivalent conditions on the asymptotics of the density of the supremum of a random walk with heavy-tailed increments. To do this, we investigate the asymptotics of the first ascending ladder height of a random walk with heavy-tailed increments. The results obtained improve and extend the corresponding classical results.
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ABSTRACT: Let S<sub>Δ</sub> denote the class of local subexponential distributions and F<sup>∗ν</sup> the ν-fold convolution of distribution F, where ν belongs to one of the following three cases: ν is a random variable taking only a finite number of values, in particular ν≡n for some n≥2; ν is a Poisson random variable; or ν is a geometric random variable. Along the lines of Embrechts, Goldie, and Veraverbeke (1979), the following assertion is proved under certain conditions: F<sup>∗ν</sup>∈S<sub>Δ</sub>⇔F∈S<sub>Δ</sub>. This result is applied to the infinitely divisible laws and some new results are established. The results obtained extend the corresponding findings of Asmussen, Foss, and Korshunov (2003).
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ABSTRACT: This paper mainly presents some global and local asymptotic estimates for the tail probabilities of the supremum and overshoot of a random walk in “the intermediate case”, where the related distributions of the increments of the random walk may not belong to the convolution equivalent distribution class. Some of the obtained results can include the classical results. For this, the paper first introduces some new distribution classes using the γ-transform of distributions, and investigates their properties and relations with some other existing distribution classes. Based on the above results, some equivalent conditions for the global and local asymptotics of the γ-transform of the distribution of the supremum of the above random walk are given. Applying these results to risk theory and infinitely divisible laws, the paper obtains some asymptotic estimates for the ruin probability and the local ruin probability of the renewal risk model with non-convolution equivalent claims, and the global and local asymptotics of an infinitely divisible law with a non-convolution equivalent Lévy measure.
Journal of Mathematical Analysis and Applications.
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ABSTRACT: This paper uses a new method to achieve some new equivalent conditions on asymptotics and local asymptotics for random sums, modifies some results based on an incorrect lemma, and cancels some technical conditions on the existing corresponding results. The newly obtained equivalent conditions are applied to risk theory and infinite divisibility theory, and some new results are derived.
Insurance: Mathematics and Economics.
