Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-23T08:56:59.371Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Properties for Increments of ${{l}^{\infty }}$-Valued Gaussian Random Fields

Published online by Cambridge University Press:  20 November 2018

Yong-Kab Choi  and
Miklós Csörgő
Yong-Kab Choi
Affiliation:
Department of Mathematics, Gyeongsang National University, Jinju 660-701, Korea e-mail: [email protected]
Miklós Csörgő
Affiliation:
School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, ON, K1S 5B6 e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper establishes general theorems which contain both moduli of continuity and large incremental results for ${{l}^{\infty }}$-valued Gaussian random fields indexed by a multidimensional parameter under explicit conditions.

Keywords

60F1560G1560G60l∞-valued Gaussian randomfieldmodulus of continuity, regularly varying functionlarge deviation probability
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 2 , 01 April 2008 , pp. 313 - 333
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Antoniadis, A. and Carmona, R., Eigenfunction expansions for infinite dimensional Ornstein-Uhlenbeck processes Probab. Theory Related Fields 74(1987), no. 1, 31–54.Google Scholar
[2] Book, S. A. and Shore, T. R., On large intervals in the Csörgʺo-Révész theorem on increments of a Wiener process Z.Wahrsch. Verw. Gebiete 46(1978), no. 1, 1–11.Google Scholar
[3] Carmona, R., Measurable norms and some Banach space valued Gaussian processes Duke Math. J. 44(1977), no. 1, 109–127.Google Scholar
[4] Choi, Y. K., Erdʺos-Rényi-type laws applied to Gaussian processes J. Math. Kyoto Univ. 31(1991), no. 3, 191–217.Google Scholar
[5] Choi, Y. K. and Hwang, K. S., How big are the lag increments of a Gaussian process? Comput. Math. Appl. 40(2000), no. 8-9, 11–919.Google Scholar
[6] Choi, Y. K. and Kôno, N., How big are the increments of a two-parameter Gaussian process? J Theoret. Probab. 12(1999), no. 1, 105–129.Google Scholar
[7] Csáki, E., Csörgʺo, M., Lin, Z. Y., and Révész, P., On infinite series of independent Ornstein-Uhlenbeck processes. Stochastic Process. Appl. 39(1991), no. 1, 25–44.Google Scholar
[8] Csörgʺo, M., Lin, Z. Y. and Shao, Q. M., Path properties for l1-valued Gaussian processes Proc. Amer. Math. Soc. 121(1994), no. 1, 225–236.Google Scholar
[9] Csörgʺo, M. and Révész, P., How big are the increments of a multi-parameterWiener process? Z.Wahrsch. Verw. Gebiete 42(1978), no. 1, 1–12.Google Scholar
[10] Csörgʺo, M. and Révész, P., Strong Approximations in Probability and Statistics . Academic Press, New York, 1981.Google Scholar
[11] Csörgʺo, M. and Shao, Q. M., Strong limit theorems for large and small increments of lp-valued Gaussian processes Ann. Probab. 21(1993), no. 4, 1958–1990.Google Scholar
[12] Csörgʺo, M. and Shao, Q. M., On almost sure limit inferior for B-valued stochastic processes and applications Probab. Theory Related Fields 99(1994), no. 1, 29–54.Google Scholar
[13] Dawson, D. A., Stochastic evolution equations. Math. Biosci. 15(1972), 287–316.Google Scholar
[14] Dawson, D. A., Stochastic evolution equations and related measure processes. J. Multivariate Anal. 5(1975), 1–52.Google Scholar
[15] Fernique, X., Continuité des processus Gaussiens. C. R. Acad. Sci. Paris 258(1964), 6058–6060.Google Scholar
[16] Gross, L., On the formula of Mathews and Salam J. Functional Analysis 25(1977), no. 2, 162–209.Google Scholar
[17] Holley, R. and Stroock, D., Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motions Publ. Res. Inst. Math. Sci 14(1978), no. 3, 741–788.Google Scholar
[18] Itô, K., Foundations of Stochastic Differential Equations in Infinite-Dimensional Spaces . CBMS-NSF Regional Conference Series in Applied Mathematics 47. Society for Industrial and Applied Mathematics(SIAM), Philadelphia, PA, 1984.Google Scholar
[19] Kôno, N., The exact modulus of continuity for Gaussian processes taking values of finite dimensional normed space . In: Trends in Probability and Related Analysis, World Scientific, River Edge, NJ, 1996, pp. 219–232.Google Scholar
[20] Kuo, H H., Gaussian Measures in Banach spaces . Lecture Notes in Mathematics 463, Springer-Verlag, Berlin, 1975.Google Scholar
[21] Leadbetter, M. R., Lindgren, G., and Rootzén, H., Extremes and Related Properties of Random Sequences and Processes . Springer-Verlag, New York, 1983.Google Scholar
[22] Li, W. V. and Shao, Q. M., A normal comparison inequality and its applications Probab. Theory Related Fields 122(2002), no. 4, 494–508.Google Scholar
[23] Lin, Z. Y. and Choi, Y. K., Some limit theorems for fractional Lévy Brownian fields Stoch. Process. Appl. 82(1999), no. 2, 229–244.Google Scholar
[24] Lin, Z. Y., Hwang, K. S., Lee, S. C. and Choi, Y. K., Path properties of a d-dimensional Gaussian process Statist. Probab. Lett. 68(2004), no. 4, 383–393.Google Scholar
[25] Lin, Z. Y. and Lu, C. R., Strong Limit Theorems . Mathematics and Its Applications (Chinese) 4, Science Press, Beijing, 1992.Google Scholar
[26] Lin, Z. Y. and Quin, Y. C., On the increments of l1-valued Gaussian processes . In: Asymptotic Methods in Probability and Statistics. North-Holland, Amsterdam, 1998, pp. 293–302.Google Scholar
[27] Ortega, J., On the size of the increments of non-stationary Gaussian processes Stoch. Process. Appl. 18(1984), no. 1, 47–56.Google Scholar
[28] Piech, M. A., The Ornstein-Uhlenbeck semigroup in an infinite dimensional L2-setting. J. Functional Analysis 18(1975), 271–285.Google Scholar
[29] Ricciardi, L. M. and Sacerdote, L., The Ornstein-Uhlenbeck process as a model for neuronal activity. Biol. Cybernetics 35(1979), 1–9.Google Scholar
[30] Schmuland, B., Some regularity results on infinite dimensional diffusions via Dirichlet forms Stochastic Anal. Appl. 6(1988), no. 3, 327–348.Google Scholar
[31] Slepian, D., The one-sided barrier problem for Gaussian noise. Bell. System Tech. J. 41(1962), 463–501.Google Scholar
[32] Steinebach, J., On a conjecture of Révész and its analogue for renewal processes . In: Asymptotic Methods in Probability and Statistics. North-Holland, Amsterdam, 1998, pp. 311–322.Google Scholar
[33] Stroock, D. W., The Malliavin calculus and its applications to second order parabolic differential equations. I Math. Syst. Theory 14(1981), no 1, 25–65.Google Scholar
[34] Walsh, J. B., A stochastic model of neural response Adv. in Appl. Probab. 13(1981), no. 2, 231–281.Google Scholar
[35] Zhang, L. X., A note on liminfs for increments of a fractional Brownian motion. Acta Math. Hungar. 76(1997), no. 1-2, 145–154.Google Scholar