From random partitions to fractional Brownian sheets. (English) Zbl 1459.60085
Summary: 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.
MSC:
| 60G22 | Fractional processes, including fractional Brownian motion |
| 60F17 | Functional limit theorems; invariance principles |
| 60G60 | Random fields |
Keywords:
fractional Brownian motion; fractional Brownian sheet; invariance principle; long-range dependence; random field; random partition; regular variationReferences:
| [1] | Bickel, P.J. and Wichura, M.J. (1971). Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Stat.42 1656-1670. · Zbl 0265.60011 · doi:10.1214/aoms/1177693164 |
| [2] | Biermé, H. and Durieu, O. (2014). Invariance principles for self-similar set-indexed random fields. Trans. Amer. Math. Soc.366 5963-5989. · Zbl 1330.60055 |
| [3] | Biermé, H., Durieu, O. and Wang, Y. (2017). Invariance principles for operator-scaling Gaussian random fields. Ann. Appl. Probab.27 1190-1234. · Zbl 1370.60057 · doi:10.1214/16-AAP1229 |
| [4] | Biermé, H., Meerschaert, M.M. and Scheffler, H.-P. (2007). Operator scaling stable random fields. Stochastic Process. Appl.117 312-332. · Zbl 1111.60033 |
| [5] | Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. New York: Wiley. · Zbl 0944.60003 |
| [6] | Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications27. Cambridge: Cambridge Univ. Press. |
| [7] | Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields. Ann. Probab.10 1047-1050. · Zbl 0496.60020 |
| [8] | Dedecker, J. (2001). Exponential inequalities and functional central limit theorems for a random fields. ESAIM Probab. Stat.5 77-104. · Zbl 1003.60033 · doi:10.1051/ps:2001103 |
| [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. · Zbl 1346.60039 · doi:10.1214/16-EJP4492 |
| [10] | Enriquez, N. (2004). A simple construction of the fractional Brownian motion. Stochastic Process. Appl.109 203-223. · Zbl 1075.60019 |
| [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. · Zbl 1189.60050 · doi:10.1214/07-PS092 |
| [12] | Hammond, A. and Sheffield, S. (2013). Power law Pólya’s urn and fractional Brownian motion. Probab. Theory Related Fields157 691-719. · Zbl 1311.60044 · doi:10.1007/s00440-012-0468-6 |
| [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. · Zbl 0982.60054 |
| [14] | Kallenberg, O. (1997). Foundations of Modern Probability. New York: Springer. · Zbl 0892.60001 |
| [15] | Kamont, A. (1996). On the fractional anisotropic Wiener field. Probab. Math. Statist.16 85-98. · Zbl 0857.60046 |
| [16] | Karlin, S. (1967). Central limit theorems for certain infinite urn schemes. J. Math. Mech.17 373-401. · Zbl 0154.43701 |
| [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. · Zbl 1075.60020 |
| [18] | Kolmogoroff, A.N. (1940). Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. Dokl. Akad. Nauk SSSR26 115-118. · JFM 66.0552.03 |
| [19] | Lavancier, F. (2007). Invariance principles for non-isotropic long memory random fields. Stat. Inference Stoch. Process.10 255-282. · Zbl 1143.62058 · doi:10.1007/s11203-006-9001-9 |
| [20] | Lei, P. and Nualart, D. (2009). A decomposition of the bifractional Brownian motion and some applications. Statist. Probab. Lett.79 619-624. · Zbl 1157.60313 · doi:10.1016/j.spl.2008.10.009 |
| [21] | Mandelbrot, B.B. and Van Ness, J.W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev.10 422-437. · Zbl 0179.47801 · doi:10.1137/1010093 |
| [22] | McLeish, D.L. (1974). Dependent central limit theorems and invariance principles. Ann. Probab.2 620-628. · Zbl 0287.60025 |
| [23] | Mikosch, T. and Samorodnitsky, G. (2007). Scaling limits for cumulative input processes. Math. Oper. Res.32 890-918. · Zbl 1279.90033 |
| [24] | Øksendal, B. and Zhang, T. (2001). Multiparameter fractional Brownian motion and quasi-linear stochastic partial differential equations. Stoch. Stoch. Rep.71 141-163. · Zbl 0986.60056 · doi:10.1080/17442500108834263 |
| [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. · Zbl 1162.60347 |
| [26] | Pipiras, V. and Taqqu, M.S. (2017). Long-Range Dependence and Self-Similarity. Cambridge Series in Statistical and Probabilistic Mathematics45. Cambridge: Cambridge Univ. Press. · Zbl 1377.60005 |
| [27] | Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Math.1875. Berlin: Springer. · Zbl 1103.60004 |
| [28] | Puplinskaitė, D. and Surgailis, D. (2015). Scaling transition for long-range dependent Gaussian random fields. Stochastic Process. Appl.125 2256-2271. · Zbl 1317.60062 · doi:10.1016/j.spa.2014.12.011 |
| [29] | Puplinskaitė, D. and Surgailis, D. (2016). Aggregation of autoregressive random fields and anisotropic long-range dependence. Bernoulli22 2401-2441. · Zbl 1356.60082 · doi:10.3150/15-BEJ733 |
| [30] | Samorodnitsky, G. (2016). Stochastic Processes and Long Range Dependence. Cham: Springer. · Zbl 1376.60007 |
| [31] | Shen, Y. and Wang, Y. (2017). Operator-scaling Gaussian random fields via aggregation. Preprint. Available at https://arxiv.org/abs/1712.07082. · Zbl 1462.60048 |
| [32] | Wang, Y. (2014). An invariance principle for fractional Brownian sheets. J. Theoret. Probab.27 1124-1139. · Zbl 1307.60030 · doi:10.1007/s10959-013-0483-2 |
| [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. · Zbl 1167.60011 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
