• LIN ZHENGYAN (Department of Mathematics Zhejiang University) ;
  • HWANG KYO-SHIN (Department of Mathematics Zhejiang University, Research Institute of Natural Science Geongsang National University)
  • Published : 2005.11.01


In this paper we establish some results on the increments of a d-dimensional Gaussian process with the usual Euclidean norm. In particular we obtain the law of iterated logarithm and the Book-Shore type theorem for the increments of ad-dimensional Gaussian process, via estimating upper bounds and lower bounds of large deviation probabilities on the suprema of the d-dimensional Gaussian process.


Gaussian process;increment;sample path behaviour


  1. M. A. Arcones, On the law of the iterated logarithm for Gaussian processes, J. Theoret. Probab. 8 (1995), no. 4, 877-903
  2. S. M. Berman, Limit theorems for the maximum term in stationary sequence, Ann. Math. Statist. 35 (1964), 502-616
  3. P. Billingsley, Probability and Measure, J. Wiley & Sons, New York, 1986
  4. S. A. Book and T. R. Shore, On large intervals in the Csorgo-Revesz theorem on increments of a Wiener process, Z. Wahrsch. verw. Gebiete 46 (1978), 1-11
  5. Y. K. Choi, Erdos-R enyi-type laws applied Gaussian processes, J. Math. Kyoto Univ. 31 (1991), no. 3, 191-217
  6. Y. K. Choi and N. Kono, How big are the increments of a two-parameter Gaussian process?, J. Theoret. Probab. 12 (1999), no. 1, 105-129
  7. E. Csaki, M. Csorgo, Z. Y. Lin, and P. Revesz, On infinite series of independent Ornstein-Uhlenbeck processes, Stochastic Process Appl. 39 (1991), 25-44
  8. E. Csaki, M. Csorgo, and Q. M. Shao, Fernique type inequalities and moduli of continuity for ${\iota}^2-valued$ Ornstein-Uhlenbeck processes, Ann. Inst. H. Poincare 28 (1992), no. 4, 479-517
  9. M. Csorgo, Z. Y. Lin, and Q. M. Shao, Path properties for $1^{\infty}$ -valued Gaussian processes, Proc. Amer. Math. Soc. 121 (1994), 225-236
  10. M. Csorgo and P. Revesz, Strong Approximations in Probability and Statistic, Academic Press, New York, 1981
  11. M. Csorgo and Q. M. Shao, Strong limit theorems for large and small increments of ${\iota}^p-valued$ Gaussian processes, Ann. Probab. 21 (1993), no. 4, 1958-1990
  12. X. Fernique, Continuite des processus Gaussiens, C. R. Math. Acad. Sci. Paris 258 (1964), 6058-6060
  13. F. X. He and B. Chen, Some results on increments of the Wiener process, Chinese J. Appl. Probab. Statist. 5 (1989), 317-326
  14. N. Kono, The exact modulus of continuity for Gaussian processes taking values of a finite dimensional normed space in: Trends in Probability and Related Analysis, SAP'96, World Scientific, Singapore, 1996, pp. 219-232
  15. Z. Y. Lin, How big the increments of a multifractional Brownian motion?, Sci. China Ser. A 45 (2002), no. 10, 1291-1300
  16. Z. Y. Lin, K. S. Hwang, S. Lee, and Y. K. Choi, Path properties of a d- dimensional Gaussian process, Statist. Probab. Lett. 68 (2004), 383-393
  17. Z. Y. Lin and C. R. Lu, Strong Limit Theorems, Science Press, Kluwer Academic Publishers, Hong Kong, 1992
  18. Z. Y. Lin and Y. C. Qin, On the increments of $1^{\infty}$-valued Gaussian processes, Asym. Methods in Probab. and Statist.(Ottawa), Elsevier, 1998, 293-302
  19. C. R. Lu, Some results on increments of Gaussian processes, Chinese J. Appl. Probab. Statist. 2 (1986), 59-65
  20. D. Monrad and H. Rootzen, Small values of Gaussian processes and functional laws of the iterated logarithm, Probab. Theory Related Fields 101 (1995), 173- 192
  21. J. Ortega, On the size of the increments of non-stationary Gaussian processes, Stochastic Process Appl. 18 (1984), 47-56
  22. Q. M. Shao, p-variation of Gaussian processes with stationary increments, Stu dia Sci. Math. Hungar. 31 (1996), 237-247
  23. L. X. Zhang, A Note on liminfs for increments of a fractional Bwownian motion, Acta Math. Hungar. 76 (1997), no. 1-2, 145-154
  24. L. X. Zhang, Some liminf results on increments of fractional Brownian motion, Acta Math. Hungar. 17 (1996), 209-234
  25. P. Revesz, A generalization of Strassen's funtional law of iterated logarithm, Z. Wahrsch. verw. Gebiete 50 (1979b), 257-264

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