Asymptotic Analysis Of Extreme Values And Related Issues

In this paper, we mainly study the asymptotic theories of extremes of random sequencesand processes and related issues from the following fve parts.Firstly, we study the extremes of a sequence under random missing. For the indepen-dently and identically distributed random sequences, the joint limit distribution and thealmost sure limit theorem between the extreme of the sequence and this extreme subtractingthe extreme of the sequence subject to random missing are obtained. The result is also ex-tended to weakly and strongly dependent Gaussian sequences. These results extend that ofMittal (1978)^[38]and Kudrov and Piterbarg (2007)^[39].Secondly, we study the almost sure limit properties of the joint version between the ex-tremes and the partial sums of stationary Gaussian sequences. Assuming that the covariancefunctions of the Gaussian sequences satisfy some conditions, the almost sure central limittheorem for the maxima (after centered at the sample mean) and the partial sums of the sta-tionary Gaussian sequence is proved. Under a very weaker condition, the almost sure centrallimit theorem for the maxima and the partial sums of the stationary Gaussian sequence isalso proved. The obtained results extend that of Dudzin′ski (2003,2008)^[57][56].Thirdly, we investigate the asymptotic relation between the extremes of stationary Gaus-sian process and the extremes of this process sampled at discrete time points. For the dis-crete time points, we distinguish it into three cases: the sparse grid points, the Pickands gridpoints and the dense grid points. If the covariance functions of the Gaussian process satisfylim^{t→∞} r(t) logt=r∈(0,∞), it is shown that the extremes of the Gaussian process andthe extremes of the sparse grid points and the Pickands grid points are asymptotic weaklydependent. If the covariance functions of the Gaussian process satisfy lim_{t→∞} r(t) logt=∞,it is proved that the extremes of the Gaussian process and the extremes of the dense gridpoints are asymptotic totally dependent. These results complete that of Piterbarg (2004)^[41].Fourthly, we study the ruin model of an aggregated Gaussian process and derive theasymptotic estimation of the fnite time ruin probability of this model and the uniform upperand lower bounds of the fnite time ruin probability. Our model includes the sub-fractionalBrownian motion, bi-fractional Brownian motion and H-self similar Gaussian martingale. We also apply our results to the classic risk model and obtain the fnite-time ruin probabilitiesof the risk processes with Gaussian perturbation. In the end, serval examples are given toclassify the importance of our main results. Our model extend that of De bicki and Sikora(2011)^[67]Finally, the almost sure limit behaviour for extremes of Gaussian felds is investigated.The almost sure central limit theorem for extremes of standardized non-stationary Gaussianfelds is proved under some conditions related to the convergence rate of covariance functions.Meanwhile, the weak convergence results of the extremes are also obtained under very weakerconditions, which improves the results of Pereira (2010)^[88].The above obtained results not only enrich the theory of extremes values theory butalso have potential applications in the felds of fnance and insurance.