Lower tail probabilities for gaussian processes

Annals of Probability

Volumn 32, Issue 1 A, 2004, Pages 216-242

  Li, Wenbo V ^a,b   Shao, Qi Man ^b  

^a UNIVERSITY OF DELAWARE   (United States)

^b UNIVERSITY OF OREGON   (United States)

Author keywords

Gaussian processes; Lower tail probability

Indexed keywords

EID: 2142769903     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1078415834     Document Type: Article

References (24)

1. BASS, R., EISENBAUM, N. and SHI, Z. (2000). The most visited sites of symmetric stable processes. Probab. Theory Related Fields 116 391-404.
2. BELLMAN, R. (1970). Introduction to Matrix Analysis. McGraw-Hill, New York. [Reprinted (1997) by SIAM.]
3. BRAMSON, M. and GRIFFEATH, D. (1991). Capture problems for coupled random walks. In Random Walks, Brownian Motion and Interacting Panicle Systems (R. Durrett and H. Kesten, eds.) 153-188. Birkhäuser, Boston.
4. CSÁKI, E., KHOSHNEVISAN, D. and SHI, Z. (2000). Boundary crossings and the distribution function of the maximum of Brownian sheet. Stochastic Process. Appl. 90 1-18.
5. DEMBO, A., POONEN, B., SHAO, Q.-M. and ZEITOUNI, O. (2002). Random polynomials having few or no real zeros. J. Amer. Math. Soc. 15 857-892.
6. DUDLEY, R. M. (1967). The sizes of compact subsets of Hubert space and continuity of Gaussian processes. J. Funct. Anal. 1 290-330.
7. FERNIQUE, X. (1964). Continuité des processus Gaussiens. C. R. Acad. Sci. Paris 258 6058-6060.
8. KESTEN, H. (1991). An absorption problem for several Brownian motions. In Seminar on Stochastic Processes (E. Çinlar, K. L. Chung and M. J. Sharpe, eds.) 59-72. Birkhäuser, Boston.
9. LEADBETTER, M. R., LINDGREN, G. and ROOTZEN, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
10. LEDOUX, M. (1996). Isoperimetry and Gaussian analysis. Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1648 165-294. Springer, New York.
11. LEDOUX, M. and TALAORAND, M. (1991). Probability on Banach Spaces. Springer, New York.
12. Li, W. V. and SHAO, Q.-M. (2001a). Gaussian processes: Inequalities, small ball probabilities and applications. In Stochastic Processes: Theory and Methods (C. R. Rao and D. Shanbhag, eds.) 533-598. North-Holland, Amsterdam.
13. Li, W. V. and SHAO, Q.-M. (2001b). Capture time of Brownian pursuits. Probab. Theory Related Fields 121 30-8.
14. Li, W. V. and SHAO, Q.-M. (2002). A normal comparison inequality and its applications. Probab. Theory Related Fields 122 494-508.
15. MARCUS, M. (2000). Probability estimates for lower levels of certain Gaussian processes with stationary increments. In High Dimensional Probability II (E. Gine, D. Mason and J. Wellner, eds.) 173-179. Birkhäuser, Boston.
16. MOLCHAN, M. (1999). Maximum of a fractional Brownian motion: Probabilities of small values. Comm. Math. Phys. 205 97-111.
17. MOLCHAN, M. (2000). On the maximum of fractional Brownian motion. Theory Probab. Appl. 44 97-102.
18. NEWELL, G. and ROSENBLATT, M. (1962). Zero crossing probabilities for Gaussian stationary processes. Ann. Math. Stat. 33 1306-1313.
19. PlCKANDS, J. (1969). Asymptotic properties of maximum in a stationary Gaussian processes. Trans. Amer. Math. Soc. 145 75-86.
20. PRICE, G. (1951). Bounds for determinants with dominant principal diagonal. Proc. Amer. Math. Soc. 2 497-502.
21. SHAO, Q.-M. (1999). A Gaussian correlation inequality and its applications to the existence of small ball constant. Preprint.
22. SLEPIAN, D. (1961). First passage time for a particular Gaussian process. Ann. Math. Statist. 32 610-612.
23. SLEPIAN, D. (1962). The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41 463-501.
24. STRAKHOV, N. and KURZ L. (1968). An upper bound on the zero-crossing distribution. Bell System Tech. J. 47 529-547.