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Sample Path Properties of Gaussian Invariant Random Fields

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Invariant Random Fields on Spaces with a Group Action

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Abstract

Let X(t) be a centred Gaussian invariant random field on a two-point homogeneous space T. The corresponding Dudley semi-metric is a function of one real variable. We use Abelian and Tauberian theorems to estimate the Dudley semi-metric and find the uniform moduli of continuity of the random field X(t). The series expansion of the multiparameter fractional Brownian motion previously obtained in Chap. 2 is shown to be rate-optimal. We prove a general functional limit theorem for the multiparameter fractional Brownian motion. The functional law of the iterated logarithm, functional Lévy modulus of continuity and many other results are its particular cases. Bibliographical remarks conclude.

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References

  • N. Aronszajn. Theory of reproducing kernels. Trans. Am. Math. Soc., 68(3):337–404, 1950. URL http://www.jstor.org/stable/1990404.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Askey. Orthogonal polynomials and special functions. Society for Industrial and Applied Mathematics, Philadelphia, 1975.

    Book  Google Scholar 

  • A. Ayache and W. Linde. Approximation of Gaussian random fields: general results and optimal wavelet representation of the Lévy fractional motion. J. Theor. Probab., 21(1):69–96, 2008. doi:10.1007/s10959-007-0101-2.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Ayache and W. Linde. Series representations of fractional Gaussian processes by trigonometric and Haar systems. Electron. J. Probab., 14(94):2691–2719, 2009 (electronic).

    MathSciNet  MATH  Google Scholar 

  • A. Ayache and M.S. Taqqu. Rate optimality of wavelet series approximations of fractional Brownian motion. J. Fourier Anal. Appl., 9(5):451–471, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  • R.M. Balan. A strong invariance principle for associated random fields. Ann. Probab., 33(2):823–840, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Baldi, G. Kerkyacharian, D. Marinucci, and D. Picard. Subsampling needlet coefficients on the sphere. Bernoulli, 15(2):438–463, 2009. URL http://projecteuclid.org/euclid.bj/1241444897.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Benassi, S. Jaffard, and D. Roux. Elliptic Gaussian random processes. Rev. Mat. Iberoam., 13(1):19–90, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Bergman. The kernel function and conformal mapping, volume 5 of Math. Surv. Am. Math. Soc., New York, 1950.

    MATH  Google Scholar 

  • S. Bergman. The kernel function and conformal mapping, volume 5 of Math. Surv. Am. Math. Soc., Providence, second, revised edition, 1970.

    MATH  Google Scholar 

  • I. Berkes and G.J. Morrow. Strong invariance principles for mixing random fields. Z. Wahrscheinlichkeitstheor. Verw. Geb., 57(1):15–37, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  • F. Biagini, Y. Hu, B. Øksendal, and T. Zhang. Stochastic calculus for fractional Brownian motion and applications. Probab. Appl. (N. Y.). Springer, London, 2008.

    Chapter  Google Scholar 

  • N.H. Bingham. Tauberian theorems for Jacobi series. Proc. Lond. Math. Soc. (3), 36:285–309, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  • N.H. Bingham. Multivariate prediction and matrix Szegö theory. Probab. Surv., 9:325–339, 2012. doi:10.1214/12-PS200.

    Article  MathSciNet  MATH  Google Scholar 

  • N.H. Bingham, C.M. Goldie, and J.L. Teugels. Regular variation, volume 27 of Encycl. Math. Appl. Cambridge University Press, Cambridge, 1987.

    MATH  Google Scholar 

  • V.I. Bogachev. Gaussian measures, volume 62 of Math. Surveys Monogr. Am. Math. Soc., Providence, 1998.

    MATH  Google Scholar 

  • W.O. Bray and M.A. Pinsky. Growth properties of Fourier transforms via moduli of continuity. J. Funct. Anal., 255:2265–2285, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Bulinski and A. Shashkin. Strong invariance principle for dependent random fields. In D. Denteneer, F. den Hollander and E. Verbitskiy, editors. Dynamics & stochastics. Festschrift in honour of M.S. Keane, volume 48 of IMS Lect. Notes Monogr. Ser., pages 128–143. Inst. Math. Statist., Beachwood, 2006.

    Chapter  Google Scholar 

  • R.M. Burton and T.-S. Kim. An invariance principle for associated random fields. Pac. J. Math., 132(1):11–19, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  • R.H. Cameron and W.T. Martin. Transformations of Wiener integrals under translations. Ann. of Math. (2), 45:386–396, 1944.

    Article  MathSciNet  MATH  Google Scholar 

  • O. Christensen. An introduction to frames and Riesz bases. Appl. Numer. Harmon. Anal. Birkhäuser, Basel, 2003.

    MATH  Google Scholar 

  • A. Dembo and O. Zeitouni. Large deviations techniques and applications, volume 38 of Appl. Math. (N. Y.). Springer, New York, second edition, 1998. Corrected reprint, 2010.

    Book  MATH  Google Scholar 

  • J.-D. Deuschel and D.W. Stroock. Large deviations, volume 137 of Pure Appl. Math. Academic Press, Boston, 1989.

    MATH  Google Scholar 

  • M.D. Donsker. An invariance principle for certain probability limit theorems. Mem. Am. Math. Soc., 1951(6):12 pp., 1951.

    MathSciNet  Google Scholar 

  • R.M. Dudley. The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal., 1:290–330, 1967.

    Article  MathSciNet  MATH  Google Scholar 

  • K. Dzhaparidze and H. van Zanten. Krein’s spectral theory and the Paley–Wiener expansion for fractional Brownian motion. Ann. Probab., 33(2):620–644, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  • K. Dzhaparidze and H. van Zanten. Optimality of an explicit series expansion of the fractional Brownian sheet. Stat. Probab. Lett., 71(4):295–301, 2006.

    Article  Google Scholar 

  • M. El Machkouri and L. Ouchti. Invariance principles for standard-normalized and self-normalized random fields. ALEA Lat. Am. J. Probab. Math. Stat., 2:177–194, 2006.

    MathSciNet  MATH  Google Scholar 

  • A. Erdélyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi. Higher transcendental functions, volume II. McGraw–Hill, New York, 1953.

    Google Scholar 

  • X. Fernique. Regularité des trajectoires des fonctions aléatoires gaussiennes. In École d’Été de Probabilités de Saint-Flour, IV–1974, volume 480 of Lect. Notes Math., pages 1–96. Springer, Berlin, 1975.

    Google Scholar 

  • M. Flensted-Jensen and T.H. Koornwinder. Positive definite spherical functions on a noncompact, rank one symmetric space. In P. Eymard, J. Faraut, G. Schiffmann, and R. Takahashi, editors. Analyse Harmonique sur les Groupes de Lie, II, volume 739 of Lect. Notes Math., pages 249–282. Springer, Berlin, 1979.

    Chapter  Google Scholar 

  • N. Gantert. An inversion of Strassen’s law of the iterated logarithm for small time. Ann. Probab., 21:1045–1049, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  • V.V. Gorodetskiĭ. The invariance principle for stationary random fields with strong mixing. Teor. Veroâtn. Ee Primen., 27(2):358–364, 1982. In Russian.

    MATH  Google Scholar 

  • V.V. Gorodetskiĭ. The central limit theorems and the invariance principle for weakly dependent random fields. Dokl. Akad. Nauk SSSR, 276(3):528–531, 1984. In Russian.

    MathSciNet  Google Scholar 

  • L. Gross. Abstract Wiener spaces. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probab. (Berkeley, CA, 1965/66), volume II, pages 31–42. University of California Press, Berkeley, 1967.

    Google Scholar 

  • H. Hedenmalm, B. Korenblum, and K. Zhu. Theory of Bergman spaces, volume 199 of Grad. Texts in Math. Springer, New York, 2000.

    Book  MATH  Google Scholar 

  • A.V. Ivanov and N.N. Leonenko. An invariance principle for an estimator of the correlation function of a homogeneous random field. Ukr. Mat. Zh., 32(3):323–331, 1980. In Russian.

    MathSciNet  MATH  Google Scholar 

  • A.V. Ivanov and N.N. Leonenko. On the invariance principle for estimating the correlation function of a uniform isotropic random field. Ukr. Mat. Zh., 33(3):313–323, 1981a. In Russian.

    MathSciNet  MATH  Google Scholar 

  • A.V. Ivanov and N.N. Leonenko. An invariance principle for an estimate of the correlation function of a homogeneous random field. Dokl. Akad. Nauk Ukrain. SSR Ser. A, 1:17–20, 1981b.

    MathSciNet  Google Scholar 

  • T.-S. Kim. The invariance principle for associated random fields. Rocky Mt. J. Math., 26(4):1443–1454, 1996.

    Article  MATH  Google Scholar 

  • T.-S. Kim and H.-Y. Seo. An invariance principle for stationary strong mixing random fields. J. Korean Stat. Soc., 24(2):281–292, 1995.

    MathSciNet  Google Scholar 

  • T.-S. Kim and H.-Y. Seo. The invariance principle for linearly positive quadrant dependent random fields. J. Korean Math. Soc., 33(4):801–811, 1996.

    MathSciNet  MATH  Google Scholar 

  • T.-S. Kim and E.-Y. Seok. The invariance principle for associated random fields. J. Korean Math. Soc., 31(4):679–689, 1994.

    MathSciNet  MATH  Google Scholar 

  • T.-S. Kim and E.-Y. Seok. The invariance principle for ρ-mixing random fields. J. Korean Math. Soc., 32(2):321–328, 1995.

    MathSciNet  MATH  Google Scholar 

  • A.N. Kolmogorov. Evaluation of minimal number of elements of ε-nets in different functional classes and their application to the problem of representation of functions of several variables by superposition of functions of a smaller number of variables. Usp. Mat. Nauk, 10(1):192–194, 1955. In Russian.

    MATH  Google Scholar 

  • A.N. Kolmogorov. On certain asymptotic characteristics of completely bounded metric spaces. Dokl. Akad. Nauk SSSR (N.S.), 108:385–388, 1956. In Russian.

    MathSciNet  MATH  Google Scholar 

  • T.H. Koornwinder. Jacobi functions and analysis on noncompact semisimple Lie groups. In Special functions: group theoretical aspects and applications, Math. Appl. pages 1–85. Reidel, Dordrecht, 1984.

    Chapter  Google Scholar 

  • Yu.V. Kozac̆enko and M.Ĭ. Yadrenko. Local properties of sample functions of random fields. I. Teor. Veroyatnost. i Mat. Statist., 14:53–66, 1976a. In Russian.

    Google Scholar 

  • Yu.V. Kozac̆enko and M.Ĭ. Yadrenko. Local properties of sample functions of random fields. II. Teor. Veroyatnost. i Mat. Statist., 15:82–98, 1976b. In Russian.

    Google Scholar 

  • T. Kühn and W. Linde. Optimal series representation of fractional Brownian sheets. Bernoulli, 8:669–696, 2002. URL http://projecteuclid.org/euclid.bj/1078435223.

    MathSciNet  MATH  Google Scholar 

  • H.S. Kuo. Gaussian measures in Banach spaces, volume 463 of Lect. Notes Math. Springer, Berlin, 1975.

    MATH  Google Scholar 

  • S. Kwapień and B. Szymański. Some remarks on Gaussian measures in Banach spaces. Probab. Math. Stat., 1(1):59–65, 1980.

    MATH  Google Scholar 

  • F. Lavancier. Invariance principles for non-isotropic long memory random fields. Stat. Inference Stoch. Process., 10(3):255–282, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  • N. Leonenko. The invariance principle for homogeneous random fields. Dopovīdī Akad. Nauk Ukraïn. RSR Ser. A, 1975(6):497–499, 1975. In Ukrainian.

    MathSciNet  MATH  Google Scholar 

  • N. Leonenko. On an invariance principle for homogeneous random fields. Teor. Veroyatnost. i Mat. Statist., 14:79–87, 1976. In Russian.

    MathSciNet  Google Scholar 

  • N. Leonenko. On the invariance principle for estimates of linear regression coefficients of a random field. Teor. Veroyatnost. i Mat. Statist., 23:80–91, 1980. In Russian.

    MathSciNet  Google Scholar 

  • N. Leonenko. On the invariance principle for estimating regression coefficients of a random field. Ukr. Mat. Zh., 33(6):771–778, 1981. In Russian.

    MathSciNet  Google Scholar 

  • N. Leonenko and M.Ĭ. Yadrenko. An invariance principle for homogeneous and isotropic random fields. Teor. Veroâtn. Ee Primen., 24(1):175–181, 1979a. In Russian.

    MathSciNet  Google Scholar 

  • N. Leonenko and M.Ĭ. Yadrenko. The invariance principle for some classes of random fields. Ukr. Mat. Zh., 31(5):559–565, 1979b. In Russian.

    MathSciNet  MATH  Google Scholar 

  • W.V. Li and Q.-M. Shao. Gaussian processes: inequalities, small probabilities, and applications. In D.N. Shanbhag and C.R. Rao, editors. Handbook of Statistics, volume 19, pages 533–597. North-Holland, Amsterdam, 2001.

    Google Scholar 

  • M.A. Lifshits. Gaussian random functions, volume 322 of Math. Appl. Kluwer Academic, Dordrecht, 1995.

    Book  Google Scholar 

  • G.G. Lorentz. Metric entropy and approximation. Bull. Am. Math. Soc., 72:903–937, 1966.

    Article  MathSciNet  MATH  Google Scholar 

  • H. Luschgy and G. Pagès. Expansions for Gaussian processes and Parseval frames. Electron. J. Probab., 14(42):1198–1221, 2009 (electronic).

    MathSciNet  MATH  Google Scholar 

  • V.E. Maiorov and G.W. Wasilkowski. Probabilistic and average linear widths in l ∞-norm with respect to r-fold Wiener measure. J. Approx. Theory, 84(1):31–40, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Malyarenko. Local properties of Gaussian random fields on compact symmetric spaces, and Jackson-type and Bernstein-type theorems. Ukr. Math. J., 51(1):66–75, 1999.

    Article  MathSciNet  Google Scholar 

  • A. Malyarenko. Abelian and Tauberian theorems for random fields on two-point homogeneous spaces. Theory Probab. Math. Stat., 69:115–127, 2005.

    Article  MathSciNet  Google Scholar 

  • A. Malyarenko. Functional limit theorems for multiparameter fractional Brownian motion. J. Theor. Probab., 19(2):263–288, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Malyarenko. An optimal series expansion of the multiparameter fractional Brownian motion. J. Theor. Probab., 21(2):459–475, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  • M.B. Marcus. A comparison of continuity conditions for Gaussian processes. Ann. Probab., 1(1):123–130, 1973a.

    Article  MATH  Google Scholar 

  • M.B. Marcus. Continuity of Gaussian processes and random Fourier series. Ann. Probab., 1:968–981, 1973b.

    Article  MATH  Google Scholar 

  • M.B. Marcus and L.A. Shepp. Continuity of Gaussian processes. Trans. Am. Math. Soc., 151:377–391, 1970.

    Article  MathSciNet  MATH  Google Scholar 

  • M.B. Marcus and L.A. Shepp. Sample behavior of Gaussian processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, CA, 1970/1971), volume II, pages 423–441. University of California Press, Berkeley, 1972.

    Google Scholar 

  • Yu.S. Mishura. Stochastic calculus for fractional Brownian motion and related processes, volume 1929 of Lect. Notes Math. Springer, Berlin, 2008.

    Book  MATH  Google Scholar 

  • D. Monrad and H. Rootzén. Small values of Gaussian processes and functional laws of the iterated logarithm. Probab. Theory Relat. Fields, 101:173–192, 1995.

    Article  MATH  Google Scholar 

  • C. Mueller. A unification of Strassen’s law and Lévy’s modulus of continuity. Z. Wahrscheinlichkeitstheor. Verw. Geb., 56:163–179, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  • C.C. Neaderhouser. An almost sure invariance principle for partial sums associated with a random field. Stoch. Process. Appl., 11(1):1–10, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  • E.J.G. Pitman. On the behavior of the characteristic function of a probability distribution in the neighborhood of the origin. J. Aust. Math. Soc., 8:423–443, 1968.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Poghosyan and S. Rœlly. Invariance principle for martingale-difference random fields. Stat. Probab. Lett., 38(3):235–245, 1998.

    Article  MATH  Google Scholar 

  • D.V. Poryvaĭ. The invariance principle for a class of dependent random fields. Theory Probab. Math. Stat., 70:123–134, 2005.

    Article  Google Scholar 

  • A.P. Prudnikov, Yu.A. Brychkov, and O.I. Marichev. Integrals and series. Vol. 1. Elementary functions. Gordon & Breach, New York, 1986.

    MATH  Google Scholar 

  • S.G. Samko, A.A. Kilbas, and O.I. Marichev. Fractional integrals and derivatives. Theory and applications. Gordon & Breach, Yverdon, 1993.

    MATH  Google Scholar 

  • G. Samorodnitsky and M.S. Taqqu. Stable non-Gaussian random processes. Stochastic models with infinite variance. Chapman & Hall, London, 1994.

    MATH  Google Scholar 

  • H. Schack. An optimal wavelet series expansion of the Riemann–Liouville process. J. Theor. Probab., 22(4):1030–1057, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Schatten and J. von Neumann. The cross-space of linear transformations. II. Ann. of Math. (2), 47:608–630, 1946. URL http://www.jstor.org/stable/1969096.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Schatten and J. von Neumann. The cross-space of linear transformations. III. Ann. of Math. (2), 49:557–582, 1948. URL http://www.jstor.org/stable/1969045.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Schilder. Some asymptotic formulas for Wiener integrals. Trans. Am. Math. Soc., 125:63–85, 1966.

    Article  MathSciNet  MATH  Google Scholar 

  • A.P. Shashkin. The invariance principle for a (bl,θ)-dependent random field. Russ. Math. Surv., 58(3):617–618, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  • A.P. Shashkin. The invariance principle for a class of weakly dependent random fields. Mosc. Univ. Math. Bull., 59(4):24–29, 2004.

    MathSciNet  Google Scholar 

  • A.P. Shashkin. A strong invariance principle for positively or negatively associated random fields. Stat. Probab. Lett., 78(14):2121–2129, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  • V. Strassen. An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheor. Verw. Geb., 3:211–226, 1964.

    Article  MathSciNet  MATH  Google Scholar 

  • V.N. Sudakov. Gauss and Cauchy measures and ε-entropy. Dokl. Akad. Nauk SSSR, 185:51–53, 1969. In Russian.

    MathSciNet  Google Scholar 

  • V.N. Sudakov. Gaussian random processes, and measures of solid angles in Hilbert space. Dokl. Akad. Nauk SSSR, 197:43–45, 1971. In Russian.

    MathSciNet  Google Scholar 

  • G. Szegö. Orthogonal polynomials, volume XXIII of Colloquium Publ. Am. Math. Soc., Providence, 1975.

    MATH  Google Scholar 

  • V.I. Tarieladze. On nuclear covariance operators. In A. Weron, editor. Proc. Second Internat. Conf., Błażejewko, 1979, volume 828 of Lect. Notes Math., pages 283–285. Springer, Berlin, 1980.

    Google Scholar 

  • N.N. Vakhania, V.I. Tarieladze, and S.A. Chobanyan. Probability distributions on Banach spaces, volume 14 of Math. Appl. (Sov. Ser.). Reidel, Dordrecht, 1987.

    Book  MATH  Google Scholar 

  • A.W. Van der Vaart and J.A. Wellner. Weak convergence and empirical processes with applications to statistics. Springer Ser. Stat. Springer, Berlin, 1996.

    Chapter  Google Scholar 

  • N.Ya. Vilenkin. Special functions and the theory of group representations, volume 22 of Transl. Math. Monogr. Am. Math. Soc., Providence, 1968. Translated from the Russian by V.N. Singh.

    MATH  Google Scholar 

  • G.N. Watson. A treatise on the theory of Bessel functions. Cambridge University Press, Cambridge, 1944. Reprinted in 1995.

    MATH  Google Scholar 

  • M.Ĭ. Yadrenko. Spectral theory of random fields. Translat. Ser. Math. Eng. Optimization Software, Publications Division, New York, 1983.

    MATH  Google Scholar 

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    Anatoliy Malyarenko

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Malyarenko, A. (2013). Sample Path Properties of Gaussian Invariant Random Fields. In: Invariant Random Fields on Spaces with a Group Action. Probability and Its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33406-1_4

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