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Frontiers of Mathematics in China

Front. Math. China    2017, Vol. 12 Issue (4) : 821-842     DOI: 10.1007/s11464-016-0620-1
RESEARCH ARTICLE |
Limit theorems for functionals of Gaussian vectors
Hongshuai DAI1, Guangjun SHEN2(), Lingtao KONG1
1. School of Statistics, Shandong University of Finance and Economics, Jinan 250014, China
2. Department of Mathematics, Anhui Normal University, Wuhu 241000, China
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Abstract

Operator self-similar processes, as an extension of self-similar processes, have been studied extensively. In this work, we study limit theorems for functionals of Gaussian vectors. Under some conditions, we determine that the limit of partial sums of functionals of a stationary Gaussian sequence of random vectors is an operator self-similar process.

Keywords Gaussian vector      operator self-similar process      operator fractional Brownian motion      scaling limit     
Corresponding Authors: Guangjun SHEN   
Issue Date: 06 July 2017
 Cite this article:   
Hongshuai DAI,Guangjun SHEN,Lingtao KONG. Limit theorems for functionals of Gaussian vectors[J]. Front. Math. China, 2017, 12(4): 821-842.
 URL:  
http://journal.hep.com.cn/fmc/EN/10.1007/s11464-016-0620-1
http://journal.hep.com.cn/fmc/EN/Y2017/V12/I4/821
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Hongshuai DAI
Guangjun SHEN
Lingtao KONG
1 ArconesM A. Limit theorems for nonlinear funcitonals of a stationary Gaussian sequence of vectors. Ann Probab, 1994, 22: 2242–2272
doi: 10.1214/aop/1176988503
2 ChungC F. Sample means, sample autocovariances, and linear regression of stationary multivariate long memory processes. Econometric Theory, 2002, 18: 51–78
doi: 10.1017/S0266466602181047
3 CohenS, MeerschaertM M, RosińskiJ. Modeling and simulation with operator scaling. Stochastic Process Appl, 2010, 120: 2390–2411
doi: 10.1016/j.spa.2010.08.002
4 DaiH. Convergence in law to operator fractional Brownian motions. J Theoret Probab, 2013, 26: 676–696
doi: 10.1007/s10959-011-0401-4
5 DavidA. Lévy Processes and Stochastic Calculus. Cambridge: Cambridge Univ Press, 2004
6 DavidsonJ, De JongR M. The functional central limit theorem and weak convergence to stochastic integrals II. Econometric Theory, 2000, 16: 643–666
doi: 10.1017/S0266466600165028
7 de HaanL. On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. Amsterdam: Math Centre, 1970
8 DidierG, PipirasV. Integral representations and properties of operator fractional Brownian motions. Bernoulli, 2011, 17: 1–33
doi: 10.3150/10-BEJ259
9 DidierG, PipirasV. Exponents, symmetry groups and classification of operator fractional Brownian motions. J Theoret Probab, 2012, 25: 353–395
doi: 10.1007/s10959-011-0348-5
10 HudsonW N, MasonJ D. Operator-self-similar processes in a finite-dimensional space. Trans Amer Math Soc, 1982, 273: 281–297
doi: 10.1090/S0002-9947-1982-0664042-7
11 LahaT L, RohatgiV K. Operator self-similar processes in ℝd.Stochastic Process Appl, 1982, 12: 73–84
doi: 10.1016/0304-4149(81)90012-0
12 LampertiL. Semi-stable stochastic processes. Trans Amer Math Soc, 1962, 104: 62–78
doi: 10.1090/S0002-9947-1962-0138128-7
13 MarinucciD, RobinsonP. Weak convergence of multivariate fractional processes. Stochastic Process Appl, 2000, 86: 103–120
doi: 10.1016/S0304-4149(99)00088-5
14 MasonJ D, XiaoY. Sample path properties of operator-self-similar Gaussian random fields. Theory Probab Appl, 2002, 46: 58–78
doi: 10.1137/S0040585X97978749
15 MeerschaertM M, SchefflerH P. Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice. New York: John Wiley and Sons, 2001
16 SamorodnitskyG, TaqquM S. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. New York: Chapman and Hall, 1994
17 Sánchez de NaranjoM V. Non-central limit theorems for non-linear functionals of k Gaussian fields. J Multivariate Anal, 1993, 44: 227–255
doi: 10.1006/jmva.1993.1013
18 Sánchez de NaranjoM V. A central limit theorem for non-linear functionals of stationary Gaussian vector processes. Statist Probab Lett, 1993, 22: 223–230
doi: 10.1016/0167-7152(94)00070-O
19 SatoK. Self-similar processes with independent increments. Probab Theory Related Fields, 1991, 89: 285–300
doi: 10.1007/BF01198788
20 TaqquM S. Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z Wahrsch Verw Gebiete, 1975, 31: 287–302
doi: 10.1007/BF00532868
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