Electronic Journal of Probability

Spectral conditions for equivalence of Gaussian random fields with stationary increments

Abolfazl Safikhani and Yimin Xiao

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This paper studies the problem of equivalence of Gaussian measures induced by Gaussian random fields (GRFs) with stationary increments and proves a sufficient condition for the equivalence in terms of the behavior of the spectral measures at infinity. The main results extend those of Stein (2004), Van Zanten (2007, 2008) and are applicable to a rich family of nonstationary space-time models with possible anisotropy behavior.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 8, 19 pp.

Received: 21 July 2018
Accepted: 22 January 2019
First available in Project Euclid: 15 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G60: Random fields
Secondary: 62H11: Directional data; spatial statistics

Gaussian random fields equivalence of measures nonstationary spectral density spatio-temporal models

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Safikhani, Abolfazl; Xiao, Yimin. Spectral conditions for equivalence of Gaussian random fields with stationary increments. Electron. J. Probab. 24 (2019), paper no. 8, 19 pp. doi:10.1214/19-EJP270. https://projecteuclid.org/euclid.ejp/1550199786

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  • [1] Ahmed Arafat, Emilio Porcu, Moreno Bevilacqua, and Jorge Mateu, Equivalence and orthogonality of gaussian measures on spheres, Journal of Multivariate Analysis 167 (2018), 306–318.
  • [2] N. Aronszajn, Theory of reproducing kernels, Transactions of the American Mathematical Society 68 (1950), 337–404.
  • [3] F. Baudoin and D. Nualart, Equivalence of Volterra processes, Stochastic Processes and Their Applications 107 (2003), no. (2), 327–350.
  • [4] Moreno Bevilacqua, Tarik Faouzi, Reinhard Furrer, and Emilio Porcu, Estimation and prediction using generalized wendland covariance functions under fixed domain asymptotics, Annals of Statistics (2018), (to appear).
  • [5] S. D. Chatterji and V. Mandrekar, Equivalence and singularity of Gaussian measures and applications, Probabilistic Analysis and Related Topics 1 (1978), 169–197.
  • [6] P. Cheridito, Mixed fractional Brownian motion, Bernoulli 7 (2001), no. (6), 913–934.
  • [7] J.-P. Chilès and P. Delfiner, Geostatistics. Modeling Spatial Uncertainty. 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2012.
  • [8] N. Cressie and H. C. Huang, Classes of nonseparable, spatio–temporal stationary covariance functions, Journal of the American Statistical Association 94 (1999), no. (448), 1330–1339.
  • [9] N.A.C. Cressie, Statistics for Spatial Data, 2nd ed., New York: Jone Wiley & Sons, 1993.
  • [10] N. Dunford and J. T. Schwartz, Linear Operators. Part 2: Spectral Theory. Self Adjoint Operators in Hilbert Space, New York, 1963.
  • [11] W. Ebeling, Functions of Several Complex Variables and Their Singularities, Vol. 83. American Mathematical Soc., 2007.
  • [12] R. Furrer, M. G. Genton, and D. Nychka, Covariance tapering for interpolation of large spatial datasets, Journal of Computational and Graphical Statistics 15 (2006), no. (3), 502–523.
  • [13] I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes: I, (Vol. 232). Springer., 1965.
  • [14] T. Gneiting, Nonseparable, stationary covariance functions for space-time data, Journal of the American Statistical Association 97 (2002), no. (458), 590–600.
  • [15] T. Gneiting, M. Genton, and P. Guttorp, Geostatistical space-time models, stationarity, separability and full symmetry, Statistical Methods for Spatio-Temporal Systems (2007), 151–175.
  • [16] P. R. Halmos, Introduction to Hilbert Space and the Theory of Spectral Multiplicity, New York: Chelsea., 1957.
  • [17] I. A. Ibragimov and I. A. Rozanov, Gaussian Random Processes, Vol. 9. New York: Springer-Verlag., 1978.
  • [18] C. G. Kaufman, M. J. Schervish, and D. W. Nychka, Covariance tapering for likelihood-based estimation in large spatial data sets, Journal of the American Statistical Association 103 (2008), no. 484, 1545–1555.
  • [19] N. Luan and Y. Xiao, Spectral conditions for strong local nondeterminism and exact Hausdorff measure of ranges of Gaussian random fields, Journal of Fourier Analysis and Applications 18 (2012), no. 1, 118–145.
  • [20] C Ma, Singularity among selfsimilar gaussian random fields with different scaling parameters and others, Stochastic Analysis and Applications (2018), (to appear).
  • [21] Vidyadhar S Mandrekar and Leszek Gawarecki, Stochastic analysis for gaussian random processes and fields: With applications, CRC Press, 2015.
  • [22] L. D. Pitt, Some problems in the spectral theory of stationary processes on $\mathbb{R} ^d$, Indiana Univ. Math. J. 23 (1973), 343–365.
  • [23] L. D. Pitt, Stationary Gaussian Markov fields on $\mathbb{R} ^d$ with a deterministic component, Journal of Multivariate Analysis 5 (1975), 300–311.
  • [24] L. I. Ronkin, Introduction to the Theory of Entire Functions of Several Variables, Vol. 44. American Mathematical Soc., 1974.
  • [25] MD Ruiz-Medina and E Porcu, Equivalence of gaussian measures of multivariate random fields, Stochastic environmental research and risk assessment 29 (2015), no. 2, 325–334.
  • [26] Abolfazl Safikhani, Nonstationary gaussian random fields with application to space and space-time modeling, Michigan State University. Statistics, 2015.
  • [27] Abolfazl Safikhani and Yimin Xiao, Covariance tapering for anisotropic nonstationary gaussian random fields with application to large scale spatial data sets, Proceedings of the 11th International Symposium on Spatial Accuracy (2014), (A. M. Shortridge, J. P. Messina, A. Finley and S. Kravchenk, editors), pp. 179–185.
  • [28] A. V. Skorokhod and M. I. Yadrenko, On absolute continuity of measures corresponding to homogeneous Gaussian fields, Theory of Probability and Its Applications 18 (1973), no. (1), 27–40.
  • [29] T. Sottinen and C. A. Tudor, On the equivalence of multiparameter Gaussian processes, Journal of Theoretical Probability 19 (2006), no. (2), 461–485.
  • [30] M. L. Stein, Interpolation of Spatial Data: Some Theory for Kriging, Springer, 1999.
  • [31] M. L. Stein, Equivalence of Gaussian measures for some nonstationary random fields, Journal of Statistical Planning and Inference 123 (2004), 1–11.
  • [32] M. L. Stein, Space-time covariance functions, Journal of the American Statistical Association 100 (2005), no. (469), 310–321.
  • [33] H. Van Zanten, When is a linear combination of independent fBm’s equivalent to a single fBm?, Stochastic Processes and Their Applications 117 (2007), 57–70.
  • [34] H. Van Zanten, A remark on the equivalence of Gaussian processes, Electronic Communications in Probability 13 (2008), 54–59.
  • [35] Y. Xiao, Strong local nondeterminism and sample path properties of Gaussian random fields, In: Asymptotic Theory in Probability and Statistics with Applications (T.-L. Lai, Q.-M. Shao and L. Qian, eds), Higher Education Press, Beijing. (2007), 136–176.
  • [36] Y. Xue, Sample path and asymptotic properties of space-time models, Thesis (Ph.D.)–Michigan State University (2011).
  • [37] Y. Xue and Y Xiao, Fractal and smoothness properties of space-time Gaussian models, Frontiers of Mathematics in China 6 (2011), 1217–1248.
  • [38] M. I. Yadrenko, Spectral Theory of Random Fields, A. V. Balakrisn’an (Ed.). Optimization Software. Publications Division., 1983.
  • [39] A. M. Yaglom, Some classes of random fields in n-dimensional space, related to stationary random processes, Theory of Probability and Its Applications 2 (1957), no. (3), 273–320.
  • [40] H. Zhang, Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics, Journal of the American Statistical Association 99 (2004), no. (465), 250–261.