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 Electronic Communications in Probability > Vol. 13 (2008) > Paper 30 open journal systems 


Exact Convergence Rate for the Maximum of Standardized Gaussian Increments

Zakhar Kabluchko, Goettingen University
Axel Munk, Goettingen University


Abstract
We prove an almost sure limit theorem on the exact convergence rate of the maximum of standardized gaussian random walk increments. This gives a more precise version of Shao's theorem ( Shao, Q.-M., 1995. On a conjecture of Révész. Proc. Amer. Math. Soc. 123, 575-582 ) in the gaussian case.


Full text: PDF

Pages: 302-310

Published on: June 17, 2008


Bibliography
  1. J. M. P. Albin On the upper and lower classes for a stationary Gaussian stochastic process. Ann. Probab. 22(1994), 77-93. Math. Review 95e:60038
  2. L. Boysen, A. Kempe, A. Munk, V. Liebscher, O. Wittich. Consistencies and rates of convergence of jump penalized least squares estimators. Ann. Statist. (2007), to appear.
  3. H. P. Chan, T. L. Lai. Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices. Ann. Probab. 34(2006), 80-121. Math. Review 2006k:60088
  4. K.L. Chung, P. Erdös, T. Sirao. On the Lipschitz's condition for Brownian motion. J. Math. Soc. Japan 11(1959), 263-274. Math. Review 22:12602
  5. P.L. Davies, A. Kovac. Local extremes, runs, strings and multiresolution. Ann. Statist. 29(2001), 1-65. Math. Review 2002c:62067
  6. P. Deheuvels, L. Devroye. Limit laws of Erdös-Rényi-Shepp type. Ann. Probab. 15(1987), 1363-1386. Math. Review 88f:60055
  7. P. Deheuvels, L. Devroye, J. Lynch. Exact convergence rate in the limit theorems of Erdös-Rényi and Shepp. Ann. Probab. 14(1986), 209-223. Math. Review 87d:60032
  8. L. Dümbgen, V.G. Spokoiny. Multiscale testing of qualitative hypotheses. Ann. Statist. 29(2001), 124-152. Math. Review 2002j:62064
  9. Z. Kabluchko. Extreme-value analysis of standardized Gaussian increments (2007). Not published. Available at http://www.arxiv.org/abs/0706.1849
  10. M.R. Leadbetter, G. Lindgren, H. Rootzén. Extremes and related properties of random sequences and processes. Springer Series in Statistics (1983). New York-Heidelberg-Berlin: Springer-Verlag. Math. Review 84h:60050
  11. P.K. Pathak, C. Qualls. A law of iterated logarithm for stationary Gaussian processes. Trans. Amer. Math. Soc. 181(1973), 185-193. Math. Review 47:9703
  12. J. Pickands. An iterated logarithm law for the maximum in a stationary Gaussian sequence. Z. Wahrscheinlichkeitstheorie verw. Geb. 12(1969), 344-353. Math. Review 40:5003
  13. P. Révész. On the increments of Wiener and related processes. Ann. Probab. 10(1982), 613-622. Math. Review 83i:60048
  14. Q.-M. Shao. On a conjecture of Révész. Proc. Amer. Math. Soc. 123(1995), 575-582. Math. Review 95c:60031
  15. D. Siegmund, E. S. Venkatraman. Using the generalized likelihood ratio statistic for sequential detection of a change-point. Ann. Statist. 23(1995), 255-271. Math. Review 96c:62135
  16. D. Siegmund, B. Yakir. Tail probabilities for the null distribution of scanning statistics. Bernoulli 6(2000), 191-213. Math. Review 2001e:62036
  17. J. Steinebach. On a conjecture of Révész and its analogue for renewal processes, in: B. Szyszkowicz, ed. Asymptotic methods in probability and statistics. A volume in honour of Miklós Csörgö. An international conference at Carleton Univ., Canada, 1997. Amsterdam: North-Holland/Elsevier. Math. Review 2000c:60033
  18. C. Qualls. The law of the iterated logarithm on arbitrary sequences for stationary Gaussian processes and Brownian motion. Ann. Probab. 5(1977), 724-739. Math. Review 56:9655
  19. C. Qualls, H. Watanabe. An asymptotic 0-1 behavior of Gaussian processes. Ann. Math. Statist. 42(1971), 2029-2035. Math. Review 46:6437
















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Electronic Communications in Probability. ISSN: 1083-589X