The Annals of Applied Probability

Invariance principles for operator-scaling Gaussian random fields

Hermine Biermé, Olivier Durieu, and Yizao Wang

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Abstract

Recently, Hammond and Sheffield [Probab. Theory Related Fields 157 (2013) 691–719] introduced a model of correlated one-dimensional random walks that scale to fractional Brownian motions with long-range dependence. In this paper, we consider a natural generalization of this model to dimension $d\geq2$. We define a $\mathbb{Z}^{d}$-indexed random field with dependence relations governed by an underlying random graph with vertices $\mathbb{Z}^{d}$, and we study the scaling limits of the partial sums of the random field over rectangular sets. An interesting phenomenon appears: depending on how fast the rectangular sets increase along different directions, different random fields arise in the limit. In particular, there is a critical regime where the limit random field is operator-scaling and inherits the full dependence structure of the discrete model, whereas in other regimes the limit random fields have at least one direction that has either invariant or independent increments, no longer reflecting the dependence structure in the discrete model. The limit random fields form a general class of operator-scaling Gaussian random fields. Their increments and path properties are investigated.

Article information

Source
Ann. Appl. Probab. Volume 27, Number 2 (2017), 1190-1234.

Dates
Received: January 2016
Revised: June 2016
First available in Project Euclid: 26 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1495764377

Digital Object Identifier
doi:10.1214/16-AAP1229

Zentralblatt MATH identifier
1370.60057

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G60: Random fields 60G18: Self-similar processes 60G22: Fractional processes, including fractional Brownian motion

Keywords
Invariance principle operator-scaling Gaussian random field long-range dependence

Citation

Biermé, Hermine; Durieu, Olivier; Wang, Yizao. Invariance principles for operator-scaling Gaussian random fields. Ann. Appl. Probab. 27 (2017), no. 2, 1190--1234. doi:10.1214/16-AAP1229. https://projecteuclid.org/euclid.aoap/1495764377


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References

  • [1] Basrak, B. and Segers, J. (2009). Regularly varying multivariate time series. Stochastic Process. Appl. 119 1055–1080.
  • [2] Benson, D. A., Meerschaert, M. M., Baeumer, B. and Scheffler, H.-P. (2006). Aquifer operator scaling and the effect on solute mixing and dispersion. Water Resources Research 42.
  • [3] Bickel, P. J. and Wichura, M. J. (1971). Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Stat. 42 1656–1670.
  • [4] Biermé, H., Benhamou, C.-L. and Richard, F. (2010). Parametric estimation for Gaussian operator scaling random fields and anisotropy analysis of bone radiograph textures. In Proceeding de la 2ème Conférence Mathématiques Pour L’image, Orléans, 2010.
  • [5] Biermé, H. and Durieu, O. (2014). Invariance principles for self-similar set-indexed random fields. Trans. Amer. Math. Soc. 366 5963–5989.
  • [6] Biermé, H., Estrade, A. and Kaj, I. (2010). Self-similar random fields and rescaled random balls models. J. Theoret. Probab. 23 1110–1141.
  • [7] Biermé, H. and Lacaux, C. (2009). Hölder regularity for operator scaling stable random fields. Stochastic Process. Appl. 119 2222–2248.
  • [8] Biermé, H., Meerschaert, M. M. and Scheffler, H.-P. (2007). Operator scaling stable random fields. Stochastic Process. Appl. 117 312–332.
  • [9] Biermé, H., Richard, F., Rachidi, M. and Benhamou, C.-L. (2009). Anisotropic texture modeling and applications to medical image analysis. In Mathematical Methods for Imaging and Inverse Problems. ESAIM Proc. 26 100–122. EDP Sci., Les Ulis.
  • [10] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [11] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge.
  • [12] Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields. Ann. Probab. 10 1047–1050.
  • [13] Clausel, M. and Vedel, B. (2011). Explicit construction of operator scaling Gaussian random fields. Fractals 19 101–111.
  • [14] Cressie, N. and Davidson, J. L. (1998). Image analysis with partially ordered Markov models. Comput. Statist. Data Anal. 29 1–26.
  • [15] Davydov, J. A. (1970). The invariance principle for stationary processes. Teor. Verojatnost. i Primenen. 15 498–509.
  • [16] Dedecker, J. (2001). Exponential inequalities and functional central limit theorems for a random fields. ESAIM Probab. Stat. 5 77–104 (electronic).
  • [17] Dedecker, J., Merlevède, F. and Peligrad, M. (2011). Invariance principles for linear processes with application to isotonic regression. Bernoulli 17 88–113.
  • [18] Deveaux, V. and Fernández, R. (2010). Partially ordered models. J. Stat. Phys. 141 476–516.
  • [19] Enriquez, N. (2004). A simple construction of the fractional Brownian motion. Stochastic Process. Appl. 109 203–223.
  • [20] Fernández, R. and Maillard, G. (2004). Chains with complete connections and one-dimensional Gibbs measures. Electron. J. Probab. 9 145–176 (electronic).
  • [21] Fernández, R. and Maillard, G. (2005). Chains with complete connections: General theory, uniqueness, loss of memory and mixing properties. J. Stat. Phys. 118 555–588.
  • [22] Hammond, A. and Sheffield, S. (2013). Power law Pólya’s urn and fractional Brownian motion. Probab. Theory Related Fields 157 691–719.
  • [23] Kamont, A. (1996). On the fractional anisotropic Wiener field. Probab. Math. Statist. 16 85–98.
  • [24] Klüppelberg, C. and Kühn, C. (2004). Fractional Brownian motion as a weak limit of Poisson shot noise processes—With applications to finance. Stochastic Process. Appl. 113 333–351.
  • [25] Kolmogoroff, A. N. (1940). Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. C. R. (Doklady) Acad. Sci. URSS (N.S.) 26 115–118.
  • [26] Lacaux, C. and Marty, R. (2011). From invariance principles to a class of multifractional fields related to fractional sheets. Preprint. Available at http://iecl.univ-lorraine.fr/~Renaud.Marty/lacaux-marty.pdf.
  • [27] Lavancier, F. (2005). Processus empirique de fonctionnelles de champs gaussiens à longue mémoire. PUB. IRMA, Lille. 63 1–26.
  • [28] Lavancier, F. (2007). Invariance principles for non-isotropic long memory random fields. Stat. Inference Stoch. Process. 10 255–282.
  • [29] Lindskog, F. (2004). Multivariate extremes and regular variation for stochastic processes. Ph.D. thesis, Dept. Mathematics, Swiss Federal Institute of Technology, Switzerland.
  • [30] Lodhia, A., Sheffield, S., Sun, X. and Watson, S. S. (2016). Fractional Gaussian fields: A survey. Probab. Surv. 13 1–56.
  • [31] Maejima, M. and Mason, J. D. (1994). Operator-self-similar stable processes. Stochastic Process. Appl. 54 139–163.
  • [32] Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422–437.
  • [33] McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. Ann. Probab. 2 620–628.
  • [34] Meerschaert, M. M. and Scheffler, H.-P. (2001). Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice. Wiley, New York.
  • [35] Mikosch, T. and Samorodnitsky, G. (2007). Scaling limits for cumulative input processes. Math. Oper. Res. 32 890–918.
  • [36] Peligrad, M. and Sethuraman, S. (2008). On fractional Brownian motion limits in one dimensional nearest-neighbor symmetric simple exclusion. ALEA Lat. Am. J. Probab. Math. Stat. 4 245–255.
  • [37] Peligrad, M. and Utev, S. (1997). Central limit theorem for linear processes. Ann. Probab. 25 443–456.
  • [38] Puplinskaitė, D. and Surgailis, D. (2015). Scaling transition for long-range dependent Gaussian random fields. Stochastic Process. Appl. 125 2256–2271.
  • [39] Puplinskaitė, D. and Surgailis, D. (2016). Aggregation of autoregressive random fields and anisotropic long-range dependence. Bernoulli 22 2401–2441.
  • [40] Resnick, S. and Greenwood, P. (1979). A bivariate stable characterization and domains of attraction. J. Multivariate Anal. 9 206–221.
  • [41] Resnick, S. and Samorodnitsky, G. (2015). Tauberian theory for multivariate regularly varying distributions with application to preferential attachment networks. Extremes 18 349–367.
  • [42] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
  • [43] Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.
  • [44] Roux, S. G., Clausel, M., Vedel, B., Jaffard, S. and Abry, P. (2013). Self-similar anisotropic texture analysis: The hyperbolic wavelet transform contribution. IEEE Trans. Image Process. 22 4353–4363.
  • [45] Samorodnitsky, G. (2006). Long range dependence. Found. Trends Stoch. Syst. 1 163–257.
  • [46] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York.
  • [47] Sheffield, S. (2007). Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 521–541.
  • [48] Spitzer, F. (1976). Principles of Random Walk, 2nd ed. Springer, New York.
  • [49] Taqqu, M. S. (1974/75). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287–302.
  • [50] Taqqu, M. S. (1986). A bibliographical guide to self-similar processes and long-range dependence. In Dependence in Probability and Statistics (Oberwolfach, 1985). Progr. Probab. Statist. 11 137–162. Birkhäuser, Boston, MA.
  • [51] Wang, Y. (2014). An invariance principle for fractional Brownian sheets. J. Theoret. Probab. 27 1124–1139.
  • [52] Xiao, Y. (2009). Sample path properties of anisotropic Gaussian random fields. In A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Math. 1962 145–212. Springer, Berlin.