Scaling transition for nonlinear random fields with long-range dependence. (English) Zbl 1373.60089
Summary: We obtain a complete description of anisotropic scaling limits and the existence of scaling transition for nonlinear functions (Appell polynomials) of stationary linear random fields on \(\mathbb{Z}^2\) with moving average coefficients decaying at possibly different rate in the horizontal and the vertical direction. The paper extends recent results on scaling transition for linear random fields in [the authors, ibid. 125, No. 6, 2256–2271 (2015; Zbl 1317.60062); Bernoulli 22, No. 4, 2401–2441 (2016; Zbl 1356.60082)].
MSC:
| 60G60 | Random fields |
| 60F05 | Central limit and other weak theorems |
| 60G18 | Self-similar stochastic processes |
| 60G22 | Fractional processes, including fractional Brownian motion |
Keywords:
scaling transition; anisotropic long-range dependence; fractionally integrated random field; Appell polynomials; multiple Itô-Wiener integral; fractional Brownian sheetReferences:
| [1] | Anh, V. V.; Leonenko, N. N.; Ruiz-Medina, M. D., Macroscaling limit theorems for filtered spatiotemporal random fields, Stoch. Anal. Appl., 31, 460-508 (2013) · Zbl 1327.62095 |
| [2] | Arcones, M. A., Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors, Ann. Probab., 22, 2242-2274 (1994) · Zbl 0839.60024 |
| [3] | Avram, F.; Taqqu, M. S., Noncentral limit theorems and Appell polynomials, Ann. Probab., 15, 767-775 (1987) · Zbl 0624.60049 |
| [4] | Bai, S.; Taqqu, M. S., Generalized Hermite processes, discrete chaos and limit theorems, Stochastic Process. Appl., 124, 1710-1739 (2014) · Zbl 1282.60043 |
| [5] | Biermé, H.; Meerschaert, M. M.; Scheffler, H. P., Operator scaling stable random fields, Stochastic Process. Appl., 117, 312-332 (2007) · Zbl 1111.60033 |
| [6] | Boissy, Y.; Bhattacharyya, B. B.; Li, X.; Richardson, G. D., Parameter estimates for fractional autoregressive spatial processes, Ann. Statist., 33, 2533-2567 (2005) · Zbl 1085.62107 |
| [7] | Bolthausen, E., On the central limit theorem for stationary mixing random fields, Ann. Probab., 10, 1047-1050 (1982) · Zbl 0496.60020 |
| [8] | Breuer, P.; Major, P., Central limit theorems for non-linear functionals of Gaussian fields, J. Multivariate Anal., 13, 425-441 (1983) · Zbl 0518.60023 |
| [9] | Bulinski, A.; Spodarev, E.; Timmermann, F., Central limit theorems for the excursion sets volumes of weakly dependent random fields, Bernoulli, 18, 100-118 (2012) · Zbl 1239.60017 |
| [10] | Cressie, N. A.C., Statistics for Spatial Data (1993), Wiley: Wiley New York · Zbl 0468.62095 |
| [11] | Dobrushin, R. L., Gaussian and their subordinated self-similar random generalized fields, Ann. Probab., 7, 1-28 (1979) · Zbl 0392.60039 |
| [12] | Dobrushin, R. L.; Major, P., Non-central limit theorems for non-linear functionals of Gaussian fields, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 50, 27-52 (1979) · Zbl 0397.60034 |
| [13] | Doukhan, P.; Lang, G.; Surgailis, D., Asymptotics of weighted empirical processes of linear random fields with long range dependence, Ann. Inst. H. Poincaré, 38, 879-896 (2002) · Zbl 1016.60059 |
| [14] | Feller, W., An Introduction to Probability Theory and Its Applications, Vol. 1 (1957), Wiley: Wiley New York · Zbl 0158.34902 |
| [15] | Gaigalas, R.; Kaj, I., Convergence of scaled renewal processes and a packet arrival model, Bernoulli, 9, 671-703 (2003) · Zbl 1043.60077 |
| [16] | Giraitis, L.; Koul, H. L.; Surgailis, D., Large Sample Inference for Long Memory Processes (2012), Imperial College Press: Imperial College Press London · Zbl 1279.62016 |
| [17] | Giraitis, L.; Surgailis, D., CLT and other limit theorems for functionals of Gaussian processes, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 70, 191-212 (1985) · Zbl 0575.60024 |
| [18] | Guo, H.; Lim, C.; Meerschaert, M., Local Whittle estimator for anisotropic random fields, J. Multivariate Anal., 100, 993-1028 (2009) · Zbl 1167.62465 |
| [19] | Hankey, A.; Stanley, H. E., Systematic application of generalized homogeneous functions to static scaling, dynamic scaling, and universality, Phys. Revi. B, 6, 3515-3542 (1972) |
| [20] | Ho, H.-C.; Hsing, T., Limit theorems for functionals of moving averages, Ann. Probab., 25, 1636-1669 (1997) · Zbl 0903.60018 |
| [21] | Hoeffding, W., Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 58, 13-30 (1963) · Zbl 0127.10602 |
| [22] | Horváth, L.; Kokoszka, P., Sample autocovariances of long-memory time series, Bernoulli, 14, 405-418 (2008) · Zbl 1155.62323 |
| [23] | Koul, H. L.; Mimoto, N.; Surgailis, D., Goodness-of-fit tests for marginal distribution of linear random fields with long memory, Metrika, 79, 165-193 (2016) · Zbl 1330.62176 |
| [24] | Lahiri, S. N.; Robinson, P. M., Central limit theorems for long range dependent spatial linear processes, Bernoulli, 22, 345-375 (2016) · Zbl 1333.60032 |
| [25] | Lavancier, F., Invariance principles for non-isotropic long memory random fields, Stat. Inference Stoch. Process., 10, 255-282 (2007) · Zbl 1143.62058 |
| [27] | Leonenko, N. N., Random Fields with Singular Spectrum (1999), Kluwer: Kluwer Dordrecht · Zbl 1001.60071 |
| [28] | Mikosch, T.; Resnick, S.; Rootzén, H.; Stegeman, A., Is network traffic approximated by stable Lévy motion or fractional Brownian motion?, Ann. Appl. Probab., 12, 23-68 (2002) · Zbl 1021.60076 |
| [29] | Neveu, J., Bases Mathématiques du Calcul de Probabilités (1964), Mason: Mason Paris · Zbl 0137.11203 |
| [30] | Nualart, D.; Peccati, G., Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab., 33, 177-193 (2005) · Zbl 1097.60007 |
| [31] | Orey, S., A central limit theorem for \(m\)-dependent random variables, Duke Math. J., 25, 543-546 (1958) · Zbl 0107.13403 |
| [32] | Phillips, P. C.B.; Moon, H. R., Linear regression limit theory for nonstationary panel data, Econometrica, 67, 1057-1111 (1999) · Zbl 1056.62532 |
| [33] | Pilipauskaitė, V.; Surgailis, D., Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes, Stochastic Process. Appl., 124, 1011-1035 (2014) · Zbl 1400.62194 |
| [34] | Pilipauskaitė, V.; Surgailis, D., Joint aggregation of random-coefficient AR(1) processes with common innovations, Statist. Probab. Lett., 101, 73-82 (2015) · Zbl 1325.62171 |
| [35] | Pilipauskaitė, V.; Surgailis, D., Anisotropic scaling of random grain model with application to network traffic, J. Appl. Probab., 53, 857-879 (2016) · Zbl 1351.60064 |
| [36] | Pratt, J. W., On interchanging limits and integrals, Ann. Math. Statist., 31, 74-77 (1960) · Zbl 0090.26802 |
| [37] | Puplinskaitė, D.; Surgailis, D., Scaling transition for long-range dependent Gaussian random fields, Stochastic Process. Appl., 125, 2256-2271 (2015) · Zbl 1317.60062 |
| [38] | Puplinskaitė, D.; Surgailis, D., Aggregation of autoregressive random fields and anisotropic long-range dependence, Bernoulli, 22, 2401-2441 (2016) · Zbl 1356.60082 |
| [39] | Skorokhod, A. V., Limit theorems for stochastic processes, Theory Probab. Appl., 1, 261-290 (1956) · Zbl 0074.33802 |
| [40] | Surgailis, D., Zones of attraction of self-similar multiple integrals, Lith. Math. J., 22, 185-201 (1982) · Zbl 0515.60057 |
| [41] | Surgailis, D., Stable limits of sums of bounded functions of long memory moving averages with finite variance, Bernoulli, 10, 327-355 (2004) · Zbl 1076.62017 |
| [42] | Taqqu, M. S., Convergence of integrated processes of arbitrary Hermite rank, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 50, 53-83 (1979) · Zbl 0397.60028 |
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