Gaussian processes: Inequalities, small ball probabilities and applications

Handbook of Statistics

Volumn 19, Issue , 2001, Pages 533-597

  Li, W V ^a   Shao, Q M ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70350344396     PISSN: 01697161     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S0169-7161(01)19019-X     Document Type: Review

References (170)

1. Anderson T.W. The integral of a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6 (1955) 170-176
2. Baldi P., and Roynette B. Some exact equivalents for the Brownian motion in Hölder norm. Prob. Theory Related Fields 93 (1992) 457-484
3. Bass R.F. Probability estimates for multiparameter Brownian processes. Ann. Prob. 16 (1988) 251-264
4. Belinsky E. Estimates of entropy numbers and Gaussian measures for classes of functions with bounded mixed derivative. J. Approx. Theory 93 (1998) 114-127
5. Berman S.M. An asymptotic bound for the tail of the distribution of the maximum of a Gaussian process with stationary increments. Ann. Inst. H. Poincaré 21 (1985) 47-57
6. (preprint). Berthet P., and Shi Z. Small ball estimates for Brownian motion under a weighted sup-norm (1998)
7. Bingham N.H. Variants on the law of the iterated logarithm. Bull. London Math. Soc. 18 (1986) 433-467
8. Bingham N.H., Goldie C.M., and Teugels J.L. Regular Variation (1987), Cambridge Univ. Press
9. p. Math. USSR-Sbornik 2 (1967) 295-317
10. Bogachev V.I. Onsager-Machlup functions for Gaussian measures. Dokl. Akad. Nauk 344 (1995) 439-441
11. Bogachev V.I. Gaussian Measures. Amer. Math. Soc. Mathematical Surveys and Monographs Vol. 62 (1998)
12. Bolthausen E. On the speed of convergence in Strassen's law of the iterated logarithm. Ann. Prob. 6 (1978) 668-672
13. Borell C. The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30 (1975) 207-216
14. Borell C. Gaussian Radon measures on locally convex spaces. Math. Scand. 38 (1976) 265-285
15. Borovkov A., and Mogulskii A. On probabilities of small deviations for stochastic processes. Siberian Advances in Mathematics 1 (1991) 39-63
16. Capitaine M. Onsager-Machlup functional for some smooth norms on Wiener space. Prob. Theory Related Fields 102 (1995) 189-201
17. Carmona R., and Nualart D. Traces of random variables on Wiener space and the Onsager-Machlup functional. J. Funct. Anal. 107 (1992) 402-438
18. Chaleyat-Maurel M., and Nualart D. Onsager-Machlup functionals for solutions of stochastic boundary value problems. Lecture Notes in Math. 1613 (1995), Springer, Cambridge 44-55
19. (preprint). Chen X., and Li W.V. Small ball probabilities for primitives of Brownian motion (1999)
20. Chung K.L. On the maximum partial sums of sequence of independent random variables. Amer. Math. Soc. 64 (1948) 205-233
21. Cox D.D. On the existence of natural rate of escape functions for infinite dimensional Bronian motions. Ann. Prob. 10 (1982) 623-638
22. Csáki E. A relation between Chung's and Strassen's law of the iterated logarithm. Z. Wahrsch. verw. Gebiete 54 (1980) 287-301
23. Csáki E. On small values of the square integral of a multiparameter Wiener process. Statistics and Probability. Proc. of the 3rd Pannonian Symp. on Math. Stat. (1982), D. Reidel Publishing Company, Berlin 19-26
24. Limit theorems in probability and statistics. Csáki E. A lim inf result in Strassen's law of the iterated logarithm. Colloquia Math. Soc. J. Bolyai 57 (1989) 83-93
25. Csáki E. Some limit theorems for empirical processes. In: Vilaplana J.P., and Puri M.L. (Eds). Proc. 4th IMSIBAC. Recent Advances in Statistics and Probability (1994), VSP 247-254
26. Csáki E., and Csörgo{double acute} M. Inequalities for increments of stochastic processes and moduli of continuity. Ann. Prob. 20 (1992) 1031-1052
27. Csörgo{double acute} M., and Horváth L. Weighted Approximations in Probability and Statistics (1993), John Wiley & Sons, Utrecht
28. Csörgo{double acute} M., and Révész P. How small are the increments of a Wiener process?. Stoch. Process Appl. 8 (1978) 119-129
29. Csörgo{double acute} M., and Révész P. Strong Approximations in Probability and Statistics (1979), Academic Press, New York
30. p-valued Gaussian processes. Ann. Prob. 21 (1993) 1958-1990
31. Csörgo{double acute} M., and Shao Q.M. On almost sure limit inferior for B-valued stochastic processes and applications. Prob. Theory Related Fields 99 (1994) 29-54
32. de Acosta A. Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm. Ann. Prob. 11 (1983) 78-101
33. Deheuvels P., and Mason D. Random fractal functional laws of the iterated logarithm. Studia Sci. Math. Hungar. 34 (1998) 89-106
34. Dembo A., Mayer-Wolf E., and Zeitouni O. Exact behavior of Gaussian seminorms. Stat. Prob. Lett. 23 (1995) 275-280
35. Dembo A., and Zeitouni O. Large deviations techniques and applications. 2nd ed. (1998), Springer, New York
36. Deuschel J., and Stroock D. Large Deviations (1989), Academic Press, New York
37. Donsker M.D., and Varadhan S.R.S. Asymptotic evaluation of certain Wiener integrals for large time. Functional integration and its applications (Proc. Interna. Conf.) (1975) 15-33
38. Donsker M.D., and Varadhan S.R.S. Asymptotics for the Wiener Sausage. Comm. Pure Appl. Math. 28 (1975) 525-565
39. Doob J.L. Heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Stat. 20 (1949) 393-403
40. Dudley R. The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1 (1967) 290-330
41. Dudley R.M., Hoffmann-Jørgensen J., and Shepp L.A. On the lower tail of Gaussian seminorms. Ann. Prob. 9 (1979) 319-342
42. Dunker T. Small ball estimates for the fractional Brownian sheet. PhD Thesis (1998), F. Schiller Universität, Boston
43. (to appear). Dunker T. Estimates for the small ball probabilities of the fractional Brownian sheet. J. Theoret. Prob. (1999)
44. Dunker T., Kuhn T., Lifshits M.A., and Linde W. Metric entropy of integration operator and small ball probabilities of the Brownian sheet. C. R. Acad. Sci. Paris 326 (1998) 347-352
45. (preprint). Dunker T., Kuhn T., Lifshits M.A., and Linde W. Approximation numbers and l-norm (1999)
46. (to appear). Dunker T., Li W.V., and Linde W. Small ball probabilities of integrals of weighted Brownian motion. Stat. Prob. Lett. (1999)
47. Birkhauser. Dunker T., Lifshits M.A., and Linde W. Small deviations of sums of independent variables. Progress in Probability 43 (1998) 59-74
48. Dvoretzky A., and Erdös P. Some problems on random walk in space. Proc. Second Berkely Symp. Math. Stat. Prob. (1951) 353-367
49. Ehrhard A. Symétrisation dans l'espace de Gauss. Math. Scand. 53 (1983) 281-301
50. Ehrhard A. Inéqualités Isopérimétriques et intégrales de Dirichlet Gaussiens. Ann. Scient Éc Norm. Sup. 17 (1984) 317-332
51. Ehrhard A. Eléments extrémaux pour les inégalités de Brunn-Minkowski gaussiennes. Ann. H. Poincaré 22 (1986) 149-168
52. Erickson K. Rates of escape of infinite dimensional Brownian motion. Ann. Prob. 8 (1980) 325-338
53. Fernique X. Régularité des trajectoires des fonctions aléatoires gaussiennes. École Été de Probabilités de St.-Flour. Lecture Notes in Math. 480 (1975) 1-96
54. Fukushima M., Oshima Y., and Takeda M. Dirichlet Forms and Symmetric Markov Processes (1994), Walter de Gruyter, Jena
55. Birkhauser. Goodman V. Some probability and entropy estimates for Gaussian measures. Progress in Probability 20 (1990) 150-156
56. Goodman V., and Kuelbs J. Rates of convergence for the functional LIL. Ann. Prob. 17 (1989) 301-316
57. Goodman V., and Kuelbs J. Rates of clustering in Strassen's LIL for Brownian motion. J. Theory Prob. 4 (1991) 285-309
58. Gordon Y. Some inequalities for Gaussian processes and applications. Israel J. Math. 50 (1985) 265-289
59. Gorn N.L., and Lifshits M.A. Chung's law and Csáki function. J. Theoret. Prob. 12 (1999) 399-420
60. Grill K. On the rate of convergence in Strassen's law of the iterated logarithm. Prob. Related Fields 74 (1987) 583-589
61. Grill K. A lim inf result in Strassen's law of the iterated logarithm. Prob. Theory Related Fields 89 (1991) 149-157
62. Grill K. Exact rate of convergence in Strassen's law of the iterated logarithm. J. Theoret. Prob. 5 (1992) 197-204
63. Gross L. Lectures in modern analysis and applications II. Lecture Notes in Math. Vol. 140 (1970), Springer, New York
64. Das Gupta S., Eaton M., Olkin I., Perlman M., Savage L., and Sobel M. Inequalities on the probability content of convex regions for elliptically contoured distributions. Proc. Sixth Berkeley Symp. Math. Stat. Prob. 3 (1972) 241-264
65. Hargé G. Une inégalité de décorrélation pour la mesure gaussienne. C. R. Acad. Sci. Paris Sér. 326 (1998) 1325-1328
66. Paper no. 2. Hitczenko P., Kwapień S., Li W.V., Schechtman G., Schlumprecht T., and Zinn J. Hypercontractivity and comparison of moments of iterated maxima and minima of independent random variables. Electronic Journal of Probability Vol. 3 (1998) 1-26
67. (in preparation). Horváth L., and Shao Q.M. Liminf for weighted empirical processes (1999)
68. Hu Y. Ito-Wiener chaos expansion with exact residual and correlation, variance inequalities. J. Theoret. Prob. 10 (1997) 835-848
69. Ibragimov I.A. On the probability that a Gaussian vector with values in a Hilbert space hits a sphere of small radius. J. Sov. Math. 20 (1982) 2164-2174
70. Jogdeo K. A simple proof of an inequality for multivariate normal probabilities of rectangles. Ann. Math. Stat. 41 (1970) 1357-1359
71. Kac M. On the average of a certain Wiener functional and a related limit theorem in calculus of probability. Trans. Amer. Math. Soc. 59 (1946) 401-414
72. Kac M. On some connections between probability theory and differential and integral equations. Proc. Second Berkeley Symposium on Math. Stat. and Probability (1951), University of California Press, Berlin 189-215
73. Khatri C.G. On certain inequalities for normal distributions and their applications to simultaneous confidence bounds. Ann. Math. Stat. 38 (1967) 1853-1867
74. (preprint). Khoshnevisan D., Peres Y., and Xiao Y. Limsup random fractals (1999)
75. Khoshnevisan D., and Shi Z. Chung's law for integrated Brownian motion. Trans. Amer. Math. Soc. 350 (1998) 4253-4264
76. Khoshnevisan D., and Shi Z. Gaussian measure of a small ball and capacity in Wiener space. Asymptotic methods in Probability and Statistics (1998), North-Holland, Berkeley 453-465
77. Kolmogorov A.N., and Tihomirov V.M. ε-entropy and ε-capacity of sets in function spaces. Amer. Math. Soc. Transl. 17 (1961) 277-364
78. Komlós J., Major P., and Tusnády G. An approximation of partial sums of independent rv's and the sample distribution function. I. Z. Wahrsch. verw. Gebiete 32 (1975) 111-131
79. Kuelbs J. A strong convergence theorem for Banach space valued random variables. Ann. Prob. 4 (1976) 744-771
80. Kuelbs J., and Li W.V. Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal. 116 (1993) 133-157
81. Kuelbs J., and Li W.V. Small ball estimates for Brownian motion and the Brownian sheet. J. Theoret. Prob. 6 (1993) 547-577
82. Kuelbs J., and Li W.V. An extension of Ehrhard's Theorem. Interaction Between Functional Analysis, Harmonic Analysis, and Probability. Lecture notes in pure and applied mathematics Vol. 175 (1995) 291-300
83. Birkhauser. Kuelbs J., and Li W.V. Some shift inequalities for Gaussian measures. Progress in Probability 43 (1998) 233-243
84. (in preparation). Kuelbs J., and Li W.V. Wichura type functional law of the iterated logarithm for fractional Brownian (2000)
85. Kuelbs J., Li W.V., and Linde W. The Gaussian measure of shifted balls. Prob. Theory Related Fields 98 (1994) 143-162
86. Kuelbs J., Li W.V., and Shao Q. Small ball estimates for fractional Brownian motion under Hölder norm and Chung's functional LIL. J. Theoret. Prob. 8 (1995) 361-386
87. Kuelbs J., Li W.V., and Talagrand M. Lim inf results for Gaussian samples and Chung's functional LIL. Ann. Prob. 22 (1994) 1879-1903
88. Kwapień S. A remark on the median and the expectation of convex functions of Gaussian vectors. Progress in Probability 35 (1994), Birkhauser, Amsterdam 271-272
89. Kwapień S., and Sawa J. On some conjecture concerning Gaussian measures of dilatations of convex symmetric sets. Studia Math. 105 (1993) 173-187
90. Lai T.L. Gaussian processes, moving averages and quick detection problems. Ann. Prob. 1 (1973) 825-837
91. Landau H.J., and Shepp L.A. On the supremum of a Gaussian process. Sankhya 32 (1970) 369-378
92. Latala R. A note on the Ehrhard's inequality. Studia Math. 118 (1996) 169-174
93. (preprint). Latala R., and Oleszkiewicz K. Gaussian measures of dilatations of convex symmetric sets (1999)
94. Ledoux M. Isoperimetry and Gaussian Analysis. Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1648 (1996), Springer-Verlag, Boston 165-294
95. Ledoux M., and Talagrand M. Probability on Banach Spaces (1991), Springer
96. Lévy P. Theorie de l'Addition des Variables Aleatoires (1937), Gauthier-Villars, Berlin
97. Lewis T., and Pritchard G. Correlation measures. Elect. Comm. Prob. 4 (1999) 77-85
98. Li W.V. Comparison results for the lower tail of Gaussian seminorms. J. Theoret. Prob. 5 (1992) 1-31
99. 2-norm. Prob. Theory and Related Fields 92 (1992) 69-90
100. p. Progress in Probability 30 (1992), Birkhauser, Paris 106-117
101. ICAMPS'97. Li W.V. A lim inf result for the Brownian motion. In: Szyszkowicz B. (Ed). Asymptotic methods in Probability and Statistics (1998), North-Holland 281-292
102. Li W.V. A Gaussian correlation inequality and its applications to small ball probabilities. Elect. Comm. Prob. 4 (1999) 111-118
103. Li W.V. Small deviations for Gaussian Markov processes under the sup-norm. J. Theoret. Prob. 12 (1999) 971-984
104. p-norm (1999)
105. (in preparation). Li W.V. The existence of small ball constants for Gaussian processes under various norms (2000)
106. Li W.V., and Linde W. Small ball problems for non-centered Gaussian measures. Prob. Math. Stat. 14 (1993) 231-251
107. Li W.V., and Linde W. Existence of small ball constants for fractional Brownian motions. C.R. Acad. Sci. Paris 326 (1998) 1329-1334
108. Li W.V., and Linde W. Approximation, metric entropy and small ball estimates for Gaussian measures. Ann. Prob. 27 (1999) 1556-1578
109. Li W.V., and Shao Q.M. Small ball estimates for Gaussian processes under the Sobolov norm. J. Theoret. Prob. 12 (1999) 699-720
110. (to appear). Li W.V., and Shao Q.M. A note on the Gaussian correlation conjecture (1999)
111. Lifshits M.A. Gaussian Random Functions (1995), Kluwer Academic Publishers, Amsterdam
112. Lifshits M. Asymptotic behavior of small ball probabilities. Probability Theory and Mathematical Statistics. TEV and VSP editions. Proceedings of Seventh Vilnius Conference (1999) 453-468
113. (preprint). Lifshits M.A., and Linde W. Entropy numbers of Volterra operators with application to Brownian motion (1999)
114. Lifshits M.A., and Tsyrelson B.S. Small ball deviations of Gaussian fields. Theoret. Prob. Appl. 31 (1986) 557-558
115. Mandelbrot B., and Van Ness J. Fractional Brownian motions, fractional noises and applications. SIAM Review 10 (1968) 422-437
116. Marcus M. Hölder conditions for continuous Gaussian processes. Trans. Amer. Math. Soc. 134 (1968) 29-52
117. Marcus M. A bound for the distribution of the maximum of continuous Gaussian processes. Ann. Math. Stat. 41 (1970) 305-309
118. Marcus M., and Shepp L. Sample behavior of Gaussian processes. Proc. Sixth Berkeley Symp. Math. Stat. Prob. 2 (1972) 423-441
119. Mogulskii A.A. Small deviations in space of trajectories. Th. Prob. Appl. 19 (1974) 726-736
120. Mogulskii A.A. On the law of the iterated logarithm in Chung's form for functional spaces. Theoret. Prob. Appl. 24 (1980) 405-413
121. Monrad D., and Rootzen H. Small values of Gaussian processes and functional laws of the iterated logarithm. Prob. Theory Related Fields 101 (1995) 173-192
122. Nisio M. On the extreme values of Gaussian processes. Osaka J. Math. 4 (1967) 313-326
123. Orey S., and Taylor S. How often on a Brownian motion path does the law of iterated logarithm fail?. Pro. Lon. Math. Soc. 28 (1974) 174-192
124. Pisier G. The Volume of Convex Bodies and Banach Space Geometry (1989), Cambridge Univ. Press, Boston
125. Piterbarg V.I. Gaussian random processes. Progress in Science and Technology. Teor. Veroyatnost. Mat. Stat. Teor Kibernet. 9 (1982) 155-198
126. Pitt L. A Gaussian correlation inequality for symmetric convex sets. Ann. Prob. 5 (1977) 470-474
127. Pitt L. Local times for Gaussian vector fields. Indiana Univ. Math. J. 27 (1978) 309-330
128. Pitt L., and Tran L.T. Local sample path properties of Gaussian fields. Ann Prob. 7 (1979) 477-493
129. Schechtman G., Schlumprecht T., and Zinn J. On the Gaussian measure of the intersection. Ann. Prob. 26 (1998) 346-357
130. Shao Q.M. A note on small ball probability of Gaussian processes with stationary increments. J. Theoret. Prob. 6 (1993) 595-602
131. Shao Q.M. Bounds and estimators of a basic constant in extreme value theory of Gaussian processes. Stat. Sinica 7 (1996) 245-257
132. (preprint). Shao Q.M. A Gaussian correlation inequality and its applications to the existence of small ball constant (1999)
133. Shao Q.M., and Wang D. Small ball probabilities of Gaussian fields. Prob. Theory Related Fields 102 (1995) 511-517
134. Shepp L., and Zeitouni O. A note on conditional exponential moments and Onsager-Machlup functionals. Ann. Prob. 20 (1992) 652-654
135. Shi Z. A generalization of the Chung-Mogulskii law of the iterated logarithm. J. Multi. Anal. 37 (1991) 269-278
136. Shorack G.R., and Wellner J.A. Empirical Processes with Applications to Statistics (1986), John Wiley & Sons, Cambridge
137. Sidak Z. Rectangular confidence regions for the means of multivariate normal distributions. J. Amer. Stat. Assoc. 62 (1967) 626-633
138. Sidak Z. On multivariate normal probabilities of rectangles: Their dependence on correlations. Ann. Math. Stat. 39 (1968) 1425-1434
139. Slepian D. The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41 (1962) 463-501
140. 2-criterion of Mises. Math. Sbornik 2 (1937) 973-993
141. Stolz W. Une méthods élémentaire pour l'évaluation de petites boules Browniennes. C.R. Acad. Sci. Paris 315 (1993) 1217-1220
142. Stolz W. Some small ball probabilities for Gaussian processes under nonuniform norms. J. Theoret. Prob. 9 (1996) 613-630
143. Strassen V. An invariance principle for the law of the iterated logarithm. Z. Wahrs. verw. Gebiete 3 (1964) 211-226
144. Sytaya G.N. On some asymptotic representation of the Gaussian measure in a Hilbert space. Theory of Stochastic Processes. Ukrainian Academy of Sciences 2 (1974) 93-104
145. Sudakov V.N. Gaussian measures, Cauchy measures and ε-entropy. Soviet Math. Dokl. 10 (1969) 310-313
146. 1978. Sudakov V.N., and Tsirelson B.S. Extremal properties of half-spaces for spherically invariant measures. translated from Zap. Nauch. Sem. L.O.M.I. 41, 14-24. J. Sov. Math. 9 (1974) 9-18
147. Szarek S., and Werner E. A nonsymmetric correlation inequality for Gaussian measure. J. Multi. Anal. 68 (1999) 193-211
148. Talagrand M. On the rate of convergence in Strassen's LIL. Progress in Probability (1992), Birkhauser, New York 339-351
149. Talagrand M. New Gaussian estimates for enlarged balls. Geometric and Funct. Anal. 3 (1993) 502-526
150. Talagrand M. The small ball problem for the Brownian sheet. Ann. Prob. 22 (1994) 1331-1354
151. Talagrand M. Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann. Prob. 23 (1995) 767-775
152. Talagrand M. Lower classes for fractional Brownian motion. J. Thoeret. Prob. 9 (1996) 191-213
153. Talagrand M. Multiple points of trajectories of multiparameter fractional Brownian motion. Prob. Theory Related Fields 112 (1998) 545-563
154. Taylor S.J. The measure theory of random fractals. Math. Proc. Cambridge Philos. Soc. 100 (1986) 383-406
155. Temlyakov V. An inequality for trigonometric polynomials and its application for estimating the entropy numbers. Journal of Complexity 11 (1995) 293-307
156. Tong Y.L. Probability Inequalities in Multi-variate Distributions (1980), Academic Press, Boston
157. Üstünel A.S. An Introduction to Analysis on Wiener Space (1995), Springer, New York
158. Varadhan S.R.S. Large Deviations and Applications, SIAM (1984), Philadelphia, Berlin
159. Vitale R. The Wills Functional and Gaussian Processes. Ann. Prob. 24 (1996) 2172-2178
160. Vitale R. Majorization and Gaussian Correlation. Stat. Prob. Lett. 45 (1999) 247-251
161. Vitale R. A log-concavity proof for a Gaussian exponential bound. Contemporary Math.: Advances in Stochastic Inequalities 234 (1999) 209-212
162. (to appear). Vitale R. Some Comparisons for Gaussian Processes. Proc. Amer. Math. Soc. (1999)
163. (unpublished). Wichura M. A functional form of Chung's Law of the iterated logarithm for maximun absolute partial sums (1973)
164. Xiao Y. Hausdorff measure of the sample paths of Gaussian random fields. Osaka J. Math. 33 (1996) 895-913
165. (preprint). Xiao Y. Fractal measures of the sets associated to Gaussian random fields (1997)
166. Xiao Y. Hausdorff-type measures of the sample paths of fractional Brownian motion. Stoch. Process. Appl. 74 (1998) 251-272
167. Yaglom A.M. Some classes of random fields in n-dimensional space, related to stationary random processes. Theory Prob. Appl. 2 (1957) 273-320
168. Zhang L.X. Two different kinds of liminfs on the LIL for two-parameter Wiener processes. Stoch. Proc. Their Appl. 63 (1996) 175-188
169. Zhang L.X. Some liminf results on increments of fractional Brownian motions. Acta Math. Hungar. 71 (1996) 215-240
170. 2. J. Sov. Math. 24 (1986) 2330-2334