高斯三角阵的聚集点过程与部分和的联合渐近行为

数学物理学报 ›› 2024, Vol. 44 ›› Issue (5): 1311-1318.

高斯三角阵的聚集点过程与部分和的联合渐近行为

鲁盈吟¹,张文静¹,郭金辉^2,^*()

¹西南石油大学理学院成都 610500
²西南财经大学统计学院成都 611130

收稿日期:2023-09-05 修回日期:2024-03-01 出版日期:2024-10-26 发布日期:2024-10-16
通讯作者: *郭金辉, E-mail: guojinhui94@163.com
基金资助:
四川省自然科学基金(2022NSFSC1838)

Joint Behavior of Point Process of Clusters and Partial Sum for a Gaussian Triangular Array

Lu Yingyin¹,Zhang Wenjing¹,Guo Jinhui^2,^*()

¹School of Science, Southwest Petroleum University, Chengdu 610500
²School of Statistics, Southwestern University of Finance and Economics, Chengdu 611130

Received:2023-09-05 Revised:2024-03-01 Online:2024-10-26 Published:2024-10-16
Supported by:
Natural Science Foundation of Sichuan(2022NSFSC1838)

摘要/Abstract

摘要：

具有单位方差的中心化平稳高斯三角阵$ \{X_{i,n},1\leq i\leq n\}$, 在其相关系数 $ \rho_{j,n}=E\left( X_{i,n}X_{i+j,n}\right)$ 满足文献 [14] 的条件下, 本文证明了该高斯三角阵的聚集点过程依分布收敛于泊松过程, 并且聚集点过程与该高斯三角阵的部分和渐近独立.

关键词: 高斯三角阵, 聚集点过程, 部分和, 渐近行为

Abstract:

Let $\{X_{i,n},1\leq i\leq n\}$ be a centered stationary Gaussian triangular array with unit variance. Assuming the correlation $ \rho_{j,n}=E\left( X_{i,n}X_{i+j,n}\right)$ satisfies the conditions in [14], this paper is interested in the joint behavior of the point process of clusters and the partial sum of the Gaussian triangular array. It is shown that the point process of clusters converges in distribution to a Poisson process and is asymptotically independent with the partial sums.

Key words: Stationary gaussian triangular array, Point process of clusters, Partial sum, Joint behavior

中图分类号:

O211.4

鲁盈吟, 张文静, 郭金辉. 高斯三角阵的聚集点过程与部分和的联合渐近行为[J]. 数学物理学报, 2024, 44(5): 1311-1318.

Lu Yingyin, Zhang Wenjing, Guo Jinhui. Joint Behavior of Point Process of Clusters and Partial Sum for a Gaussian Triangular Array[J]. Acta mathematica scientia,Series A, 2024, 44(5): 1311-1318.

参考文献 27

[1]	Anderson C W, Turkman K F. The joint limiting distribution of sums and maxima of stationary sequences. Journal of Applied Probability, 1991, 28(1): 33-34 doi: 10.2307/3214738
[2]	Anderson C W, Turkman K F. Limiting joint distributions of sums and maxima in a statistical context. Theory of Probability and its Applications, 1993, 37(2): 314-316
[3]	Anderson C W, Turkman K F. Sums and maxima of stationary sequences with heavy tailed distributions. Sankhya Series A, 1995, 57: 1-10
[4]	Berman S M. Limit theorems for the maximum term in stationary sequences. The Annals of Mathematical Statistics, 1964, 35(2): 502-516
[5]	Biondi F, Kozubowski T J, Panorska A K. A new model for quantifying climate episodes. International Journal of Climatology, 2005, 25(9): 1253-1264
[6]	Buitendag S, Beirlant J, De Wet T. Confidence intervals for extreme Pareto-type quantiles. Scandinavian Journal of Statistics, 2020, 47(1): 36-55
[7]	Chow T L, Teugels J L. The sum and the maximum of iid random variables. North-Holland: Amsterdam, 1978, 45: 394-403
[8]	French J P, Davis R A. The asymptotic distribution of the maxima of a Gaussian random field on a lattice. Extremes, 2013, 16: 1-26
[9]	Hashorva E, Peng L, Weng Z. Maxima of a triangular array of multivariate Gaussian sequence. Statistics and Probability Letters, 2015, 103: 62-72
[10]	Hashorva E, Weng Z. Limit laws for extremes of dependent stationary Gaussian arrays. Statistics and Probability Letters, 2013, 83(1): 320-330
[11]	Hill B M. A simple general approach to inference about the tail of a distribution. The Annals of Statistics, 1975, 3: 1163-1174
[12]	Ho H C, Hsing T. On the asymptotic joint distribution of the sum and maximum of stationary normal random variables. Journal of Applied Probability, 1996, 33(1): 138-145
[13]	Ho H C, McCormick W P. Asymptotic distribution of sum and maximum for strongly dependent Gaussian processes. Journal of Applied Probability, 1999, 36: 1031-1044
[14]	Hsing T, Hüsler J, Reiss R D. The extremes of a trangular array of normal random variables. The Annals of Applied Probability, 1996, 6(2): 671-686
[15]	Hu A, Peng Z, Qi Y. Joint behavior of point process of exceedances and partial sum from a Gaussian sequence. Metrika, 2009, 70(3): 279-295
[16]	James B, James K, Qi Y. Limit distribution of the sum and maximum from multivariate Gaussian sequences. Journal of Multivariate Analysis, 2007, 98(3): 517-532
[17]	Kallenberge O. Random Measures. Berlin: Akademie-Verlag, 1976
[18]	Kozubowski T J, Panorska A K. A mixed bivariate distribution with exponential and geometric marginals. Journal of Statistical Planning and Inference, 2005, 134(2): 501-520
[19]	Kozubowski T J, Panorska A K, Qeadan F. A new multivariate model involving geometric sums and maxima of exponentials. Journal of Statistical Planning and Inference, 2011, 141(7): 2353-2367
[20]	Leadbetter M R. Extremes and local dependence in stationary sequences. Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete, 1983, 65(2): 291-306
[21]	Leadbetter M R, Lindgren G, Rootzen H. Extremes and Related Properties of Stationary Sequences and Processes. Berlin: Springer-Verlag, 1983
[22]	Ling C. Extremes of stationary random fields on a lattice. Extremes, 2019, 22: 391-411 doi: 10.1007/s10687-019-00349-z
[23]	McCormick W P, Qi Y. Asymptotic distribution for the sum and maximum of Gaussian processes. Journal of Applied Probability, 2000, 37(4): 958-971
[24]	Mittal Y, Ylvisaker D. Limit distribution for the maximum of stationary Gaussian processes. Stochastic Processes and their Applications, 1975, 3(1): 1-18
[25]	O'Brien G L. Extreme values for stationary and Markov sequences. Annals of Probability, 1987, 15(1): 281-291
[26]	Peng Z, Nadarajah S. On the joint limiting distribution of sums and maxima of stationary normal sequence. Theory of Probability and Its Applications, 2003, 47(4): 706-709
[27]	谭中权, 彭作祥. 部分和乘积的几乎处处中心极限定理. 数学物理学报, 2009, 29A(6): 1689-1698
	Tan Z Q, Peng Z X. Almost sure central limit theorem for the product of partial sums. Acta Math Sci, 2009, 29A(6): 1689-1698