## Bernoulli

• Bernoulli
• Volume 25, Number 2 (2019), 1412-1450.

### From random partitions to fractional Brownian sheets

#### Abstract

We propose discrete random-field models that are based on random partitions of $\mathbb{N}^{2}$. The covariance structure of each random field is determined by the underlying random partition. Functional central limit theorems are established for the proposed models, and fractional Brownian sheets, with full range of Hurst indices, arise in the limit. Our models could be viewed as discrete analogues of fractional Brownian sheets, in the same spirit that the simple random walk is the discrete analogue of the Brownian motion.

#### Article information

Source
Bernoulli, Volume 25, Number 2 (2019), 1412-1450.

Dates
Revised: January 2018
First available in Project Euclid: 6 March 2019

https://projecteuclid.org/euclid.bj/1551862855

Digital Object Identifier
doi:10.3150/18-BEJ1025

#### Citation

Durieu, Olivier; Wang, Yizao. From random partitions to fractional Brownian sheets. Bernoulli 25 (2019), no. 2, 1412--1450. doi:10.3150/18-BEJ1025. https://projecteuclid.org/euclid.bj/1551862855

#### References

• [1] Bickel, P.J. and Wichura, M.J. (1971). Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Stat. 42 1656–1670.
• [2] Biermé, H. and Durieu, O. (2014). Invariance principles for self-similar set-indexed random fields. Trans. Amer. Math. Soc. 366 5963–5989.
• [3] Biermé, H., Durieu, O. and Wang, Y. (2017). Invariance principles for operator-scaling Gaussian random fields. Ann. Appl. Probab. 27 1190–1234.
• [4] Biermé, H., Meerschaert, M.M. and Scheffler, H.-P. (2007). Operator scaling stable random fields. Stochastic Process. Appl. 117 312–332.
• [5] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. New York: Wiley.
• [6] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge: Cambridge Univ. Press.
• [7] Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields. Ann. Probab. 10 1047–1050.
• [8] Dedecker, J. (2001). Exponential inequalities and functional central limit theorems for a random fields. ESAIM Probab. Stat. 5 77–104.
• [9] Durieu, O. and Wang, Y. (2016). From infinite urn schemes to decompositions of self-similar Gaussian processes. Electron. J. Probab. 21 Paper No. 43, 23.
• [10] Enriquez, N. (2004). A simple construction of the fractional Brownian motion. Stochastic Process. Appl. 109 203–223.
• [11] Gnedin, A., Hansen, B. and Pitman, J. (2007). Notes on the occupancy problem with infinitely many boxes: General asymptotics and power laws. Probab. Surv. 4 146–171.
• [12] Hammond, A. and Sheffield, S. (2013). Power law Pólya’s urn and fractional Brownian motion. Probab. Theory Related Fields 157 691–719.
• [13] Hu, Y., Øksendal, B. and Zhang, T. (2000). Stochastic partial differential equations driven by multiparameter fractional white noise. In Stochastic Processes, Physics and Geometry: New Interplays, II (Leipzig, 1999). CMS Conf. Proc. 29 327–337. Providence, RI: Amer. Math. Soc.
• [14] Kallenberg, O. (1997). Foundations of Modern Probability. New York: Springer.
• [15] Kamont, A. (1996). On the fractional anisotropic Wiener field. Probab. Math. Statist. 16 85–98.
• [16] Karlin, S. (1967). Central limit theorems for certain infinite urn schemes. J. Math. Mech. 17 373–401.
• [17] 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.
• [18] Kolmogoroff, A.N. (1940). Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. Dokl. Akad. Nauk SSSR 26 115–118.
• [19] Lavancier, F. (2007). Invariance principles for non-isotropic long memory random fields. Stat. Inference Stoch. Process. 10 255–282.
• [20] Lei, P. and Nualart, D. (2009). A decomposition of the bifractional Brownian motion and some applications. Statist. Probab. Lett. 79 619–624.
• [21] Mandelbrot, B.B. and Van Ness, J.W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422–437.
• [22] McLeish, D.L. (1974). Dependent central limit theorems and invariance principles. Ann. Probab. 2 620–628.
• [23] Mikosch, T. and Samorodnitsky, G. (2007). Scaling limits for cumulative input processes. Math. Oper. Res. 32 890–918.
• [24] Øksendal, B. and Zhang, T. (2001). Multiparameter fractional Brownian motion and quasi-linear stochastic partial differential equations. Stoch. Stoch. Rep. 71 141–163.
• [25] 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.
• [26] Pipiras, V. and Taqqu, M.S. (2017). Long-Range Dependence and Self-Similarity. Cambridge Series in Statistical and Probabilistic Mathematics 45. Cambridge: Cambridge Univ. Press.
• [27] Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Math. 1875. Berlin: Springer.
• [28] Puplinskaitė, D. and Surgailis, D. (2015). Scaling transition for long-range dependent Gaussian random fields. Stochastic Process. Appl. 125 2256–2271.
• [29] Puplinskaitė, D. and Surgailis, D. (2016). Aggregation of autoregressive random fields and anisotropic long-range dependence. Bernoulli 22 2401–2441.
• [30] Samorodnitsky, G. (2016). Stochastic Processes and Long Range Dependence. Cham: Springer.
• [31] Shen, Y. and Wang, Y. (2017). Operator-scaling Gaussian random fields via aggregation. Preprint. Available at https://arxiv.org/abs/1712.07082.
• [32] Wang, Y. (2014). An invariance principle for fractional Brownian sheets. J. Theoret. Probab. 27 1124–1139.
• [33] 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. Berlin: Springer.