Let be random processes defined as the differences of two independent stationary chi-type processes with and degrees of freedom. In this paper we derive the asymptotics of under some assumptions on the covariance structures of the underlying Gaussian processes. Further, we establish a Berman sojourn limit theorem and a Gumbel limit result.
Mots-clés : Stationary Gaussian process, stationary chi-type process, extremes, Berman sojourn limit theorem, Gumbel limit theorem, Berman’s condition
@article{PS_2016__20__349_0, author = {Albin, Patrik and Hashorva, Enkelejd and Ji, Lanpeng and Ling, Chengxiu}, title = {Extremes and limit theorems for difference of chi-type processes}, journal = {ESAIM: Probability and Statistics}, pages = {349--366}, publisher = {EDP-Sciences}, volume = {20}, year = {2016}, doi = {10.1051/ps/2016018}, zbl = {1356.60042}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ps/2016018/} }
TY - JOUR AU - Albin, Patrik AU - Hashorva, Enkelejd AU - Ji, Lanpeng AU - Ling, Chengxiu TI - Extremes and limit theorems for difference of chi-type processes JO - ESAIM: Probability and Statistics PY - 2016 SP - 349 EP - 366 VL - 20 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2016018/ DO - 10.1051/ps/2016018 LA - en ID - PS_2016__20__349_0 ER -
%0 Journal Article %A Albin, Patrik %A Hashorva, Enkelejd %A Ji, Lanpeng %A Ling, Chengxiu %T Extremes and limit theorems for difference of chi-type processes %J ESAIM: Probability and Statistics %D 2016 %P 349-366 %V 20 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2016018/ %R 10.1051/ps/2016018 %G en %F PS_2016__20__349_0
Albin, Patrik; Hashorva, Enkelejd; Ji, Lanpeng; Ling, Chengxiu. Extremes and limit theorems for difference of chi-type processes. ESAIM: Probability and Statistics, Tome 20 (2016), pp. 349-366. doi : 10.1051/ps/2016018. http://archive.numdam.org/articles/10.1051/ps/2016018/
J.M.P. Albin, On extremal theory for non differentiable stationary processes. Ph.D. thesis, University of Lund, Sweden (1987).
On extremal theory for stationary processes. Ann. Probab. 18 (1990) 92–128. | Zbl
,On a test statistic for linear trend. Extremes 6 (2003) 247–258. | DOI | Zbl
and ,Extreme value theory for stochastic integrals of Legendre polynomials. J. Multivariate Anal. 100 (2009) 1029–1043. | DOI | Zbl
, and ,Sojourns and extremes of stationary processes. Ann. Probab. 10 (1982) 1–46. | DOI | Zbl
,S.M. Berman, Sojourns and extremes of stochastic processes. The Wadsworth and Brooks/Cole Statistics/Probability Series. Wadsworth and Brooks/Cole Advanced Books and Software, Pacific Grove, CA (1992). | Zbl
Gaussian approximation of perturbed chi-square risks. Stat. Interface 7 (2014) 363–373. | DOI | Zbl
, and ,On asymptotic constants in the theory of Gaussian processes. Bernoulli 20 (2014) 1600–1619. | DOI | Zbl
and ,P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling extremal events for insurance and finance. Springer-Verlag, Berlin (1997). | Zbl
Piterbarg theorems for chi-processes with trend. Extremes 18 (2015) 37–64. | DOI | Zbl
and ,Asymptotic expansion of Gaussian chaos via probabilistic approach. Extremes 18 (2015) 315–347. | DOI | Zbl
, and ,On convergence of the uniform norms for Gaussian processes and linear approximation problems. Ann. Appl. Probab. 13 (2003) 1615–1653. | DOI | Zbl
, and ,Detecting non-simultaneous changes in means of vectors. TEST 24 (2015) 681–700. | DOI | Zbl
,Outcrossings of safe regions by generalized hyperbolic processes. Stat. Probab. Lett. 83 (2013) 2197–2204. | DOI | Zbl
and ,M.R. Leadbetter, G. Lindgren and H. Rootzén, Vol. 11 of Extremes and related properties of random sequences and processes. Springer Verlag (1983). | Zbl
Extreme value theory for continuous parameter stationary processes. Z. Wahrsch. Verw. Gebiete 60 (1982) 1–20. | DOI | Zbl
and ,Extreme values and crossings for the -process and other functions of multidimensional Gaussian processes, with reliability applications. Adv. Appl. Probab. 12 (1980) 746–774. | Zbl
,Extremal ranks and transformation of variables for extremes of functions of multivariate Gaussian processes. Stochastic Process. Appl. 17 (1984) 285–312. | DOI | Zbl
,Slepian models for -processes with dependent components with application to envelope upcrossings. J. Appl. Probab. 26 (1989) 36–49. | Zbl
,On maxima of chi-processes over threshold dependent grids. Statistics 50 (2016) 579–595. | DOI | Zbl
and ,Extremes of order statistics of self-similar processes (in Chinese). Sci. Sin. Math. 46 (2016) 1–10. DOI: 10.1360/012016-15.
and ,V.I. Piterbarg, Asymptotic methods in the theory of Gaussian processes and fields, Vol. 148. American Mathematical Society, Providence, RI (1996). | Zbl
On probability of high extremes for product of two independent Gaussian stationary processes. Extremes 18 (2015) 99–108. | DOI | Zbl
and ,Asymptotic behavior of mean uniform norms for sequences of Gaussian processes and fields. Extremes 8 (2006) 161–169 (2005). | DOI | Zbl
,Exact asymptotics and limit theorems for supremum of stationary -processes over a random interval. Stochastic Process. Appl. 123 (2013) 1983–2998. | Zbl
and ,Limit theorems for extremes of strongly dependent cyclo-stationary -processes. Extremes 16 (2013) 241–254. | DOI | Zbl
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