# zbMATH — the first resource for mathematics

Self-similar random fields and rescaled random balls models. (English) Zbl 1213.60096
Author’s abstract: We study generalized random fields which arise as rescaling limits of spatial configurations of uniformly scattered random balls as the mean radius of the balls tends to 0 or infinity. Assuming that the radius distribution has a power-law behavior, we prove that the centered and renormalized random balls field admits a limit with self-similarity properties. Our main result states that all self-similar, translation- and rotation-invariant Gaussian fields can be obtained through a unified zooming procedure starting from a random balls model. This approach has to be understood as a microscopic description of macroscopic properties. Under specific assumptions, we also get a Poisson-type asymptotic field. In addition to investigating stationarity and self-similarity properties, we give $$L ^{2}$$-representations of the asymptotic generalized random fields viewed as continuous random linear functionals.

##### MSC:
 60G60 Random fields 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G18 Self-similar stochastic processes 60D05 Geometric probability and stochastic geometry 60G20 Generalized stochastic processes 60F05 Central limit and other weak theorems
Full Text:
##### References:
 [1] Adler, R.J.: The Geometry of Random Field. Wiley, New York (1981) · Zbl 0478.60059 [2] Benassi, A., Jaffard, S., Roux, D.: Elliptic Gaussian random processes. Rev. Mat. Iberoam. 13(1), 19–89 (1997) · Zbl 0880.60053 [3] Biermé, H.: Champs aléatoires: autosimilarité, anisotropie et étude directionnelle. PhD report. http://www.math-info.univ-paris5.fr/$$\sim$$bierme/recherche/Thesehb.pdf (2005) [4] Biermé, H., Estrade, A.: Poisson random balls: self-similarity and X-ray images. Adv. Appl. Probab. 38(1), 1–20 (2006) · Zbl 1094.60004 · doi:10.1239/aap/1143936137 [5] Biermé, H., Estrade, A., Kaj, I.: About scaling behavior of random balls models. In: S 4 G 6th Int. Conference, pp. 63–68. Union of Czech Mathematicians and Physicists, Prague (2006) [6] Chi, Z.: Construction of stationary self-similar generalized fields by random wavelet expansion. Probab. Theory Relat. Fields 121, 269–300 (2001) · Zbl 1008.60055 · doi:10.1007/PL00008805 [7] Cioczek-Georges, R., Mandelbrot, B.B.: A class of micropulses and antipersistent fractional Brownian motion. Stoch. Process. Their Appl. 60, 1–18 (1995) · Zbl 0846.60055 · doi:10.1016/0304-4149(95)00046-1 [8] Cioczek-Georges, R., Mandelbrot, B.B.: Alternative micropulses and fractional Brownian motion. Stoch. Process. Their Appl. 64, 143–152 (1996) · Zbl 0879.60076 · doi:10.1016/S0304-4149(96)00089-0 [9] Dobrushin, R.L.: Gaussian and their subordinated self-similar random generalized fields. Ann. Probab. 7(1), 1–28 (1979) · Zbl 0392.60039 · doi:10.1214/aop/1176995145 [10] Guelfand, I.M., Chilov, G.E.: Les Distributions I. Dunod, Paris (1962) · Zbl 0115.10102 [11] Guelfand, I.M., Vilenkin, N.Y.: Les Distributions IV: Applications de l’Analyse Harmonique. Dunod, Paris (1967) [12] Guérin, C.A.: Wavelet analysis and covariance structure of non-stationary processes. J. Fourier Anal. Appl. 6, 403–425 (2000) · Zbl 0958.60034 · doi:10.1007/BF02510146 [13] Herbin, E.: From N-parameter fractional Brownian motions to N-parameter multifractional Brownian motion. Rocky Mt. J. Math. 36(4), 1249–1284 (2006) · Zbl 1135.60020 · doi:10.1216/rmjm/1181069415 [14] Kaj, I.: Limiting fractal random processes in heavy-tailed systems. In: Fractals in Engineering, New Trends in Theory and Applications, pp. 199–218. Springer, London (2005) · Zbl 1186.60018 [15] Kaj, I., Leskelä, L., Norros, I., Schmidt, V.: Scaling limits for random fields with long-range dependence. Ann. Probab. 35, 528–550 (2007) · Zbl 1134.60027 · doi:10.1214/009117906000000700 [16] Kaj, I., Taqqu, M.S.: Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach. In: Vares, M.E., Sidoravicius, V. (eds.) Out of Equilibrium 2. Progress in Probability, vol. 60, pp. 383–427. Birkhäuser, Basel (2008) · Zbl 1154.60020 [17] Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, Berlin (2002) · Zbl 0996.60001 [18] Matheron, G.: The intrinsic random functions and their applications. Adv. Appl. Probab. 5, 439–468 (1973) · Zbl 0324.60036 · doi:10.2307/1425829 [19] Medina, J.M., Cernuschi-Frías, B.: A synthesis of 1/f process via Sobolev spaces and fractional integration. IEEE Trans. Inf. Theory 51(12), 4278–4285 (2005) · Zbl 1317.60008 · doi:10.1109/TIT.2005.858933 [20] Perrin, E., Harba, R., Berzin-Joseph, C., Iribarren, I., Bonami, A.: nth-order fractional Brownian motion and fractional Gaussian noises. IEEE Trans. Signal Process. 45, 1049–1059 (2001) · doi:10.1109/78.917808 [21] Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1966) · Zbl 0142.01701 [22] Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman & Hall, London (1994) · Zbl 0925.60027 [23] Takenaka, S.: Integral-geometric construction of self-similar stable processes. Nagoya Math. J. 123, 1–12 (1991) · Zbl 0757.60035 [24] Yaglom, A.M.: Correlation Theory of Stationary and Related Random Functions (I). Springer, Berlin (1997)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.