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Zhongquan Tan

Publications and source records attributed to Zhongquan Tan.

14 recordsLinked to original sources
arXiv2024

On the extreme order statistics for stationary Gaussian sequences subject to random missing observations

Let $\mathbf{X}=\{X_{n}\}_{n\geq 1}$ be a sequence of stationary Gaussian variables and suppose that only some of the random variables from $\mathbf{X}$ can be observed. In this paper, by studying the limiting properties of multidimensional exceedance point processes for $\mathbf{X}$, we derived the joint limit distribution of extreme order statistics for the Gaussian sequence $\mathbf{X}$ and its observed ones. The joint limit distribution of the locations and heights of the maxima for the Gaussian sequence $\mathbf{X}$ and its observed ones are also obtained.

math.PR↗
arXiv2023

On the maxima of nonstationary random fields subject to missing observations

Motivated by the papers of Mladenovc and Piterbarg (2006), Krajka (2011) and Pereira and Tan (2017), we study the limit properties for the maxima from nonstationary random fields subject to missing observations and obtain the weakly convergence and almost sure convergence results for these maxima. Some examples such as Gaussian random fields, $chi$-random fields and Gaussian order statistics fields are given to illustrate the obtained results.

math.PR↗
arXiv2023

The asymptotic distribution of maxima of stationary random sequences under random replacing

In this paper, we investigated the effect on extreme of random replacing for a stationary sequence satisfying a type of long dependent condition and a local dependent condition, and derived the joint asymptotic distribution of maximum from the stationary sequence and the maximum from the random replacing sequence. We also provided several applications for our main results.

math.PR↗
arXiv2023

The asymptotic relation between the first crossing point and the last exit time of Gaussian order statistics sequences

In this paper, we study the asymptotic relation between the first crossing point and the last exit time for Gaussian order statistics which are generated by stationary weakly and strongly dependent Gaussian sequences. It is shown that the first crossing point and the last exit time are asymptotically independent and dependent for weakly and strongly dependent respectively. The asymptotic relation between the first exit time and the last exit time for stationary weakly and strongly dependent Gaussian Gaussian sequences are also obtained.

math.PR↗
arXiv2020

On the distributional expansions of powered extremes from Maxwell distribution

In this paper, asymptotic expansions of the distributions and densities of powered extremes for Maxwell samples are considered. The results show that the convergence speeds of normalized partial maxima relies on the powered index. Additionally, compared with previous result, the convergence rate of the distribution of powered extreme from Maxwell samples is faster than that of its extreme. Finally, numerical analysis is conducted to illustrate our findings.

math.PR↗
arXiv2019

On the maxima of continuous and discrete time Gaussian order statistics processes

In this paper, we study the asymptotic relation between the maximum of acontinuous order statistics process formed by stationary Gaussian processesand the maximum of this process sampled at discrete time points. It is shown that, these two maxima are asymptotically independent when the Gaussian processes are weakly dependent and the discrete points are sufficient sparse, while for other case, these two maxima are asymptotically dependent.

math.PR↗
arXiv2019

Extremes of a type of locally stationary Gaussian random fields with applications to Shepp statistics

Let $\{Z(τ,s), (τ,s)\in [a,b]\times[0,T]\}$ with some positive constants $a,b,T$ be a centered Gaussian random field with variance function $σ^{2}(τ,s)$ satisfying $σ^{2}(τ,s)=σ^{2}(τ)$. We firstly derive the exact tail asymptotics for the maximum $M_{H}(T)=\max_{(τ,s)\in[a,b]\times[0,T]}Z(τ,s)/σ(τ)$ up crossing some level $u$ with any fixed $0 0$; and we further derive the extreme limit law for $M_{H}(T)$. As applications of the main results, we derive the exact tail asymptotics and the extreme limit law for Shepp statistics with stationary Gaussian process, fractional Brownian motion and Gaussian integrated process as input.

math.PR↗
arXiv2017

Limit laws on extremes of non-homogeneous Gaussian random fields

In this paper, by using the exact tail asymptotics derived by Debicki, Hashorva and Ji (Ann. Probab. 2014), we proved the Gumbel limit theorem for the maximum of a class of non-homogeneous Gaussian random fields. By using the obtained results, we also derived the Gumbel laws for Shepp statistics of fractional Brownian motion and Gaussian integrated process as well as the Gumbel law for Storage process with fractional Brownian motion as input.

math.PR↗
arXiv2015

On maxima of chi-processes over threshold dependent grids

In this paper, with motivation from [30] by Piterbarg (Extremes 7:161--177, 2004) and the considerable interest in stationary chi-processes, we derive asymptotic joint distributions of maxima of stationary strongly dependent chi-processes on a continuous time and an uniform grid on the real axis. Our findings extend those for Gaussian cases and give three involved dependence structures via the strongly dependence condition and the sparse, Pickands and dense grids.

math.PR↗
arXiv2015

On Piterbarg's max-discretisation theorem for homogeneous Gaussian random fields

Motivated by the papers of Piterbarg (2004) and Hüsler (2004), in this paper the asymptotic relation between the maximum of a continuous dependent homogeneous Gaussian random field and the maximum of this field sampled at discrete time points is studied. It is shown that, for the weakly dependent case, these two maxima are asymptotically independent, dependent and coincide when the grid of the discrete time points is a sparse grid, Pickands grid and dense grid, respectively, while for the strongly dependent case, these two maxima are asymptotically totally dependent if the grid of the discrete time points is sufficiently dense, and asymptotically dependent if the the grid points are sparse or Pickands grids.

math.PR↗
arXiv2014

Finite-time ruin probability of aggregate Gaussian processes

Let $\left\{\sum_{i=1}^n λ_i X_i(t), t\in [0,T]\right\}$ be an aggregate Gaussian risk process with $X_i, i\leq n$ independent Gaussian processes satisfying Piterbarg conditions and $λ_i$'s given positive weights. In this paper we derive exact asymptotics of the finite-time ruin probability given by $$\mathbb{P}\left(\sup_{t\in[0,T]}\left(\sum_{i=1}^n λ_i X_i(t)- g(t) \right)>u\right)$$ as $u\to\infty$ for some general trend function $g$. Further, we derive asymptotic results for the finite-time ruin probabilities of risk processes perturbed by an aggregate Gaussian process.

math.PR↗
arXiv2013

Large Deviations of Shepp Statistics for Fractional Brownian Motion

Define the incremental fractional Brownian field $B_{H}(s+τ)-B_{H}(s), H\in (0,1)$, where $B_{H}(s)$ is a standard fractional Brownian motion with Hurst index $H\in(0,1)$. In this paper we derive the exact asymptotic behaviour of the maximum $\max_{(τ,s)\in[0,1]\times[0,T]} (B_{H}(s+τ)-B_{H}(s)) $ for any $H\in (0,1/2)$ complimenting thus the result of Zholud (2008) for the Brownian motion.

math.PR↗