Advertisement

Journal of Theoretical Probability

, Volume 18, Issue 1, pp 79–97 | Cite as

Asymptotics and Bounds for Multivariate Gaussian Tails

  • Enkelejd HashorvaEmail author
Article

Let {X n , n ≥ 1} be a sequence of centered Gaussian random vectors in \({\mathbb R}^{d}\) , d ≥ 2. In this paper we obtain asymptotic expansions (n → ∞) of the tail probability P{X n >t n } with t n ɛ \({\mathbb R}^{d}\) a threshold with at least one component tending to infinity. Upper and lower bounds for this tail probability and asymptotics of discrete boundary crossings of Brownian Bridge are further discussed.

Keywords

Tail asymptotics Gaussian random sequences Discrete boundary crossing Brownian bridge Quadratic programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bischoff, W., Hashorva, E., Hüsler, J., and Miller, F. (2002). Asymptotically optimal test for a change-point regression problem with application in quality control. Manuscript.Google Scholar
  2. Bischoff, W., Hashorva, E., Hüsler, J., Miller, F. 2003aAsymptotics of a boundary crossing probability of a Brownian bridge with general trendMethodol. Comp. Appl. Probab.5271287Google Scholar
  3. Bischoff, W., Hashorva, E., Hüsler, J., Miller, F. 2003bExact asymptotics for boundary crossings of the Brownian bridge with trend with application to the Kolmogorov testAnn. Inst. Statist. Math.55849864Google Scholar
  4. Bischoff, W., Hashorva, E., Hüsler, J., Miller, F. 2004On the power of the Kolmogorov test to detect the trend of a Brownian bridge with applications to a change-point problem in regression modelsStat. Prob. Lett.66105115Google Scholar
  5. Dai, M., Mukherjea, A. 2001Identification of the parameters of a multivariate normal vector by the distribution of the minimumJ. Theoret. Prob.14267298Google Scholar
  6. Elnaggar, M., Mukherjea, A. 1999Identification of the parameters of a trivariate normal vector by the distribution of the minimumJ. Statist. Plann. Inference782337Google Scholar
  7. Gjacjauskas, E. 1973Estimation of the multidimensional normal probability distribution law for a receding hyperangle, LitovskMat. Sb.138390Google Scholar
  8. Hashorva, E., Hüsler, J. 2002aOn asymptotics of multivariate integrals with applications to recordsStochastic Models184169Google Scholar
  9. Hashorva, E., Hüsler, J. 2002bRemarks on compound poisson approximation of gaussian random sequencesStatist. Prob. Lett.5718Google Scholar
  10. Hashorva, E., Hüsler, J. 2003On multivariate gaussian tailsAnn. Inst. Statist. Math.55507522Google Scholar
  11. Kallenberg, O. 1997Foundations of Modern ProbabilitySpringerNew YorkGoogle Scholar
  12. Mukherjea, A., Stephens, R. 1990The problem of identification of parameters by the distribution of the maximum random variable: solution for the trivariate normal caseJ. Multivariate Analysis3495115Google Scholar
  13. Raab, M. 1999Compound Poisson approximation of the number of exceedances in Gaussian sequencesExtremes1295321Google Scholar
  14. Satish, I. 1986On a lower bound for the multivariate normal Mills’ ratioAnn. Probab.1413991403Google Scholar
  15. Savage, I. R. 1962Mills’ ratio for multivariate normal distributionJ. Res. Nat. Bur. Standards Sect. B669396Google Scholar
  16. Steck, G. P. 1979Lower bounds for the multivariate normal Mills’ ratioAnn. Probab.7547551Google Scholar
  17. Tong, Y. L. 1989The Multivariate Normal DistributionSpringerBerlinGoogle Scholar
  18. Wlodzimierz, B. 1995Normal Distribution: Characterizations with Applications. Lecture Notes in Statistics. 100SpringerBerlinGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of BernBernSwitzerland

Personalised recommendations