DOI QR코드

DOI QR Code

LIMIT BEHAVIORS FOR THE INCREMENTS OF A d-DIMENSIONAL MULTI-PARAMETER GAUSSIAN PROCESS

  • CHOI YONG-KAB (Department of Mathematics Gyeongsang National University) ;
  • LIN ZRENGYAN (Department of Mathematics Zhejiang University) ;
  • SUNG HWA-SANG (Department of Mathematics Gyeongsang National University) ;
  • HWANG KYO-SHIN (Department of Mathematics Gyeongsang National University) ;
  • MOON HEE-JIN (Department of Mathematics Gyeongsang National University)
  • Published : 2005.11.01

Abstract

In this paper, we establish limit theorems containing both the moduli of continuity and the large incremental results for finite dimensional Gaussian processes with N parameters, via estimating upper bounds of large deviation probabilities on suprema of the Gaussian processes.

Keywords

Gaussian process;quasi-increasing;regularly varying function;large deviation probability

References

  1. Y. K. Choi, Erdos-Renyi-type laws applied to Gaussian processes, J. Math. Kyoto Univ. 31 (1991), no. 3, 191-217 https://doi.org/10.1215/kjm/1250519901
  2. Y. K. Choi, Asymptotic behaviors for the increments of Gaussian random fields, J. Math. Anal. Appl. 246 (2000), no. 2, 557-575 https://doi.org/10.1006/jmaa.2000.6818
  3. Y. K. Choi and K. S. Hwang, How big are the lag increments of a Gaussian process?, Comput. Math. Appl. 40 (2000), 911-919 https://doi.org/10.1016/S0898-1221(00)85002-6
  4. 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 https://doi.org/10.1023/A:1021796610843
  5. 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 https://doi.org/10.1016/0304-4149(91)90029-C
  6. M. Csorgo, Z. Y. Lin, and Q. M. Shao, On moduli of continuity for local time of Gaussian processes, Stochastic Process Appl. 58 (1995), 1-21 https://doi.org/10.1016/0304-4149(94)00012-I
  7. M. Csorgo, R. Norvaisa, and B. Szyszkowicz, Convergence of weighted partial sums when the limiting distribution is not necessarily random, Stochastic Process Appl. 81 (1999), 81-101 https://doi.org/10.1016/S0304-4149(98)00100-8
  8. M. Csorgo and P. Revesz, How big are the increments of a multi-parameter Wiener process?, Z. Wahrsch. verw. Gebiete 42 (1978), 1-12 https://doi.org/10.1007/BF00534203
  9. M. Csorgo and P. Revesz, Strong Approximations in Probability and Statistics, Academic Press, New York, 1981
  10. M. Csorgo and Q. M. Shao, Strong limit theorems for large and small increments of $l^p$-valued Gaussian processes, Ann. Probab. 21 (1993), no. 4, 1958-1990 https://doi.org/10.1214/aop/1176989007
  11. M. Csorgo and Q. M. Shao, On almost sure limit inferior for B-valued stochastic processes and applications, Probab. Theory Related Fields 99 (1994), 29-54 https://doi.org/10.1007/BF01199589
  12. M. Csorgo, B. Szyszkowicz, and Q. Wang, Donsker's theorem for self-normalized partial sum processes, Ann. Probab. 31 (2003), 1228-1240 https://doi.org/10.1214/aop/1055425777
  13. P. Deheuvels and J. Steinebach, Exact convergence rates in strong approximation laws for large increments of partial sums, Probab. Theory Related Fields 76 (1987), 369-393 https://doi.org/10.1007/BF01297492
  14. N. Kono, The exact modulus of continuity for Gaussian processes taking values of finite dimensional normed space, Trends in Probab. Related Analysis, SAP'96, World Scientific, Singapore, 1996, 219-232
  15. M. R. Leadbetter, G. Lindgren, and H. Rootzen, Extremes and Related Properties of Random Sequences and Processes, Springer-Verlag, New York, 1983
  16. Z. Y. Lin and Y. K. Choi, Some limit theorems for fractional Levy Brownian fields, Stochastic Process Appl. 82 (1999), 229-244 https://doi.org/10.1016/S0304-4149(99)00019-8
  17. Z. Y. Lin and C. R. Lu, Strong Limit Theorems, Kluwer Academic Publ., Hong Kong, 1992
  18. Z. Y. Lin and Y. K. Choi, Some limit theorems on the increments of a multi-parameter fractional Brownian motion, Stochastic Anal. Appl. 19 (2001), no. 4, 499-517 https://doi.org/10.1081/SAP-100002099
  19. Z. Y. Lin and Y. C. Qin, On the increments of $l^{\infty}$-valued Gaussian processes, Asymptotic Methods in Probab. and Statistics edited by B. Szyszkowicz, Elsevier Science, 1998, 293-302
  20. J. Ortega, On the size of the increments of non-stationary Gaussian processes, Stochastic Process Appl. 18 (1984), 47-56 https://doi.org/10.1016/0304-4149(84)90160-1
  21. Q. M. Shao, Recent developments on self-normalized limit theorems, Asymptotic Methods in Probability and Statistics edited by B. Szyszkowicz, Elsevier Science, 1998, 467-480
  22. D. Slepian, The one-sided barrier problem for Gaussian noise, Bell. System Tech. J. 41 (1962), 463-501 https://doi.org/10.1002/j.1538-7305.1962.tb02419.x
  23. J. Steinebach, On a conjecture of Revesz and its analogue for renewal processes, Asymptotic Methods in Probability and Statistics edited by B. Szyszkowicz, Elsevier Science, 1998, 311-322
  24. B. Szyszkowicz, $L_p-approximations$ of weighted partial sum processes, Stochastic Process Appl. 45 (1993), 295-308 https://doi.org/10.1016/0304-4149(93)90076-G
  25. L. X. Zhang, Some liminf results on increments of fractional Brownian motions, Acta Math. Hungar. 71 (1996), no. 3, 215-240 https://doi.org/10.1007/BF00052111
  26. L. X. Zhang, A note on liminfs for increments of a fractional Brownian motion, Acta Math. Hungar. 76 (1997), 145-154 https://doi.org/10.1007/BF02907058
  27. W. V. Li and Q. M. Shao, A normal comparison inequality and its application, Probab. Theory Related Fields 122 (2002), 494-508 https://doi.org/10.1007/s004400100176
  28. J. Steinebach, On the increments of partial sum processes with multi-dimensional indices, Z. Wahrsch. verw. Gebiete 63 (1983), 59-70 https://doi.org/10.1007/BF00534177

Cited by

  1. Path properties of l p -valued Gaussian random fields vol.50, pp.10, 2007, https://doi.org/10.1007/s11425-007-0084-6