arXiv subjects

Zhaolei Cui

Publications and source records attributed to Zhaolei Cui.

9 recordsLinked to original sources

arXiv2025

On the random-time and finite-time ruin probability for widely dependent claim sizes and inter-arrival times

Using the results of precise large deviation and renewal theory for widely dependent random variables, this paper obtains the asymptotic estimation of the random-time ruin probability and the uniform asymptotic estimation of finite-time ruin probability for a nonstandard renewal risk model, in which both claim sizes and the inter-arrival times of claim sizes are widely dependent.

math.PR↗

arXiv2024

Precise large deviations of some risk objectives related to the net loss process in two nonstandard risk models

For two nonstandard renewal risk models, we investigate the precise large deviations of the finite-time ruin probability and a random sum of the net-loss process, and the asymptotics of the random-time ruin probability. Notably, in one of these models, claim sizes series and claim interval time series are allowed to be arbitrarily dependent. Subsequently, we apply these results to obtain precise large deviations of proportional-net-loss process and excess-of-net-loss process, as well as asymptotic estimates of the mean of stop-net-loss reinsurance treaty. These results all involve the income items of the risk model, which are relatively rare in the existing references.

math.PR↗

arXiv2024

A Breiman's theorem for conditional dependent random vector and its applications to risk theory

In this paper, we give a Breiman's theorem for conditional dependent random vector, where one component has a regularly-varying-tailed distribution with the index $α\ge0$ and its slowly varying function satisfies a relaxed condition, while the other component is non-negative and its tail distribution is lighter than the former. This result substantially extends and improves Theorem 2.1 of Yang and Wang (Extremes,\ 2013). %with a lower moment condition requirement for many occasions. We also provide some concrete examples and some interesting properties of conditional dependent random vector. Further, we apply the above Breiman's theorem to risk theory, and obtain two asymptotic estimates of the finite-time ruin probability and the infinite-time ruin probability of a discrete-time risk model, in which the corresponding net loss and random discount are conditionally dependent.

math.PR↗

arXiv2021

Precise large deviations of sums of widely dependent random variables and its applications

In this paper, we obtain some results on precise large deviations for non-random and random sums of widely dependent random variables with common dominatedly varying tail distribution or consistently varying tail distribution on $(-\infty,\infty)$. Then we apply the results to reinsurance and insurance and give some asymptotic estimates on proportional reinsurance, random-time ruin probability and the finite-time ruin probability.

math.PR↗

arXiv2019

On the long tail property of product convolution

Let $X$ and $Y$ be two independent random variables with corresponding distributions $F$ and $G$ supported on $[0,\infty)$. The distribution of the product $XY$, which is called the product convolution of $F$ and $G$, is denoted by $H$. In this paper, some suitable conditions about $F $ and $G $ are given, under which the distribution $H$ belongs to the long-tailed distribution class. Here, $F$ is a generalized long-tailed distribution and is not necessarily an exponential distribution. Finally, a series of examples are given to show that the above conditions are satisfied by many distributions and one of them is necessary in some sense.

math.PR↗

arXiv2018

On the almost decrease of a subexponential density

For a subexponential density, so far, there has been no positive conclusion or counter example to show whether it is almost decreasing. In this paper, a subexponential density supported on $\mathbb{R}^+\cup\{0\}$ without the almost decrease is constructed by a little skillful method. The density is a positive piecewise linear function with a more normal shape. Correspondingly, there exists a local subexponential distribution which is not locally almost decreasing. Based on an example of Cline \cite{C1986}, some similar results are also obtained for the long-tailed density excluding the subexponential density and the local long-tailed distribution excluding the local subexponential distribution. Finally, the paper shows that, for the local subexponentiality of a distribution supported on $\mathbb{R}$, the local almost decreasing condition is necessary in some sense.

math.PR↗

arXiv2018

Asymptotics of convolution with the semi-regular-variation tail and its application to risk

In this paper, according to a certain criterion, we divide the exponential distribution class into three subclasses. One of them is closely related to the regular-variation-tailed distribution class, so it is called the semi-regular-variation-tailed distribution class. In the class, although all distributions are not convolution equivalent,they still have some good properties. We give the precise tail asymptotic expression of convolution of these distributions, and prove that the new class is closed under convolution. In addition, we do not need to require the corresponding random variables to be identically distributed. Finally, we apply these results to a discrete time risk model with stochastic returns, and obtain the precise asymptotic estimation of the finite time ruin probability.

math.PR↗

arXiv2017

Some positive conclusions related to the Embrechts-Goldie' conjecture

In this paper, we give some conditions, under which, if an infinitely divisible distribution supported on $[0,\infty)$ belongs to the intersection of exponential distribution class $\mathcal{L}(γ)$ for some $γ\ge0$ and generalised subexponential distribution class $\mathcal{OS}$, then its L$\rm\acute{e}$vy spectral distribution or convolution of the distribution with itself also belongs to the same one. To this end, we discuss the closure under the compound convolution roots for the class. In addition, we do some in-depth discussion about the above-mentioned conditions, and provide some types of distributions satisfying them. Further, we obtain some local versions of the above-mentioned results by the Esscher transform of distributions. Therefore, some positive conclusions related to the Embrechts-Goldie conjecture are obtained. Prior to this, all corresponding results are negative

math.PR↗

arXiv2015

On the structure of a class of distributions obeying the principle of a single big jump

In this paper, we present several heavy-tailed distributions belonging to the new class J of distributions obeying the principle of a single big jump introduced by Beck et al. [1]. We describe the structure of this class from different angles. First, we show that heavy-tailed distributions in the class J are automatically strongly heavy-tailed and thus have tails which are not too irregular. Second, we show that such distributions are not necessarily weakly tail equivalent to a subexponential distribution. We also show that the class of heavy-tailed distributions in J which are neither long-tailed nor dominatedly-varying-tailed is not only non-empty but even quite rich in the sense that it has a nonempty intersection with several other well-established classes. In addition, the integrated tail distribution of some particular of these distributions shows that the Pakes-Veraverbeke-Embrechts Theorem for the class J in [1] does not hold trivially.

math.PR↗