Summary
A criterion on almost sure limit inferior for the increments of B-valued stochastic processes is presented. Applications to processes of independent increments and to Gaussian processes with stationary increments are given. In particular, an exact limit inferior bound is established for increments of infinite series of independent Ornstein-Uhlenbeck processes.
Article PDF
Similar content being viewed by others
References
Adler, R.J.: An introduction to continuity, Extrema, and related topics for Gaussian processes (Lecture Notes Math., vol. 12) Berlin Heidelberg New York: Springer 1990
Csáki, E., Csörgő, M.: Fernique type inequalities for not necessarily Gaussian processes. Ann. Probab.20, 1031–1052 (1992)
Csáki, E., Csörgő, M., Lin, Z.Y., Révész, P.: On infinite series of independent Ornstein-Uhlenbeck processes. Stochastic Process. Appl.39, 25–44 (1991)
Csáki, E., Csörgő, M., Shao, Q.-M.: Fernique type inequalities and moduli of continuity for ℓ2-valued Ornstein-Uhlenbeck processes. Ann. Inst. Henri Poincaré Probab. Statist.28, 479–517 (1992)
Csáki, E., Csörgő, M., Shao, Q.-M.: Moduli of continuity for ℓp-valued Gaussian processes. Tech. Rep. Ser. Res. Stat. Probab. No. 160. Ottawa: Carleton University, University of Ottawa 1991
Csörgő, M., Horváth, L., Shao, Q.-M.: Convergence of integrals of uniform empirical and quantile processes. Stochastic Process. Appl.45, 283–294 (1993)
Csörgő, M., Révész, P.: Strong approximations in probability and statistics. Budapest: Akadémiai Kiadó; New York: Academic Press 1981
Csörgő, M., Shao, Q.-M.: Strong limit theorems for large and small increments of ℓp-valued Gaussian processes. Ann. Probab.21, 1958–1990 (1993)
Dawson, D.A.: Stochastic evolution equations. Math. Biosci.15, 287–316 (1972)
Dawson, D.A.: Stochastic evolution equations and related measure processes. J. Multivariate Anal.5, 1–52 (1975)
Fernique, X.: Continuite des processus Gaussiens. C.R. Acad. Sci., Paris258, 6058–6060 (1964)
Fernique, X.: La regularité des fonctions aléatoires d'Ornstein-Uhlenbeck dans ℓ2; le cas diagonal. C.R. Acad. Sci., Paris309, 59–62 (1989)
Fernique, X.: Sur la régularité de certaines fonctions aléatoires d'Ornstein-Uhlenbeck. Ann. Inst. Henri Poincaré, Probab. Stat.26, 399–417 (1990)
Fernique, X.: Regularité de fonctions aléatoires gaussiennes stationnaires. Probab. Theory Relat. Fields88, 521–536 (1991)
Fernique, X.: Sur la régularité des fonctions aléatoires d'Ornstein-Uhlenbeck à valeurs dans ℓ p ,p∈[1, ∞]. Ann. Probab.20, 1441–1449 (1992)
Iscoe, I., McDonald, D.: Large deviations for ℓ2-valued Ornstein-Uhlenbeck processes. Ann. Probab.17, 58–73 (1989)
Iscoe, I., Marcus, M., McDonald, D., Talagrand, M., Zinn, J.: Continuity of ℓp-valued Ornstein-Uhlenbeck processes. Ann. Probab.18, 68–84 (1990)
Jain, N.C., Pruitt, W.E.: Maxima of partial sums of independent random variables. Z. Wahrscheinlichkeitstheorie Verw. Geb.27, 145–151 (1973)
Jain, N.C.: A Donsker-Varadhan type of invariance principle. Z. Wahrscheinlichkeitstheorie Verw. Geb.59, 117–138 (1982)
Khatri, C.G.: On certain inequatilites for normal distribution and their applications to simultaneous confidence bounds. Ann. Math. Stat.38, 1853–1867 (1967)
Ledoux, M., Talagrand, M.: Probability in Banach spaces. Berlin Heidelberg New York: Springer 1991
Pitt, L.D., Tran, L.T.: Local sample path properties of Gaussian fields, Ann. Probab.7, 477–493 (1979)
Schmuland, B.: Dirichlet Forms and Infinite Dimensional Ornstein-Uhlenbeck Processes. Ph.D. Dissertation. Ottawa: Carleton University 1987
Schmuland, B.: Some regularity results on infinite dimensional diffusions via Dirichlet forms. Stochastic Anal. Appl.6, 327–348 (1988)
Schmuland, B.: Sample path properties of ℓ2-valued Ornstein-Uhlenbeck processes. Can. Math. Bull.33, 358–366 (1990)
Shao, Q.-M.: A note on small ball probability of Gaussian process with stationary increments. J. Theor. Probab.6, 595–602 (1993)
Taylor, S.J.: Sample path properties of a transient stable process. J. Math. Mech.16, 1229–1246 (1967)
Tong, Y.L.: Probability Inequalities in Multivariate Distributions. New York: Academic Press 1980
Walsh, J.B.: A stochastic model of neural response. Adv. Appl. Probab.13, 231–281 (1981)
Wiener, N.: Differential space. J. Math. Phys.2, 132–174 (1923)
Author information
Authors and Affiliations
Additional information
Work supported by an NSERC Canada grant at Carleton University
Work supported by the Fok Yingtung Education Foundation of China
Rights and permissions
About this article
Cite this article
Csörgő, M., Shao, QM. On almost sure limit inferior for B-valued stochastic processes and applications. Probab. Th. Rel. Fields 99, 29–54 (1994). https://doi.org/10.1007/BF01199589
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01199589