Abstract
The results of this paper concern rates of convergence for increments of Brownian motion. As a by-product we give some improvements of a result of Bolthausen dealing with Strassen's law of the iterated logarithm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Goodman, V. (1986). Characteristics of normal samples, to appearAnn. Probability.
Acosta, A. de, and Kuelbs, J. (1983). Limit theorems for moving averages of independent random vectors.Z. Wahrsch. verw. Gebiete 64, 67–123.
Bolthausen, E. (1978). On the speed of convergence in Strassen's law of the iterated logarithm.Ann. Probability 6, 668–672.
Acosta, A. de (1983). Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm.Ann. Probability 11, 78–101.
Csörgö, M., and Révész, P. (1981).Strong Approximations in Probability and Statistics, Academic, New York.
Kuelbs, J., and LePage, R. (1973). The law of the iterated logarithm for Brownian motion in a Banch space.Trans. Am. Math. Soc. 185, 253–264.
Goodman, V., Kuelbs, J., and Zinn, J. (1981). Some results on the LIL in Banach space with applications to weighted empirical processes.Ann. Probability 9, 713–752.
Kuelbs, J. (1976). A strong convergence theorem for Banach space valued random variables.Ann. Probability 4, 744–771.
Hoffmann-Jorgensen, J., Shepp, L. A., and Dudley, R. M. (1979). On the lower tail of Gaussian semi-norms.Ann. Probability 7, 319–342.
Strassen, V. (1964). An invariance principle for the law of the iterated logarithm.Z. Wahrsch. verw. Gebiete 3, 211–226.
Breiman, L. (1967). On the tail behavior of sums of independent random variables.Z. Wahrsch. verw. Gebiete 9, 20–25.
Talagrand, M. (1984). Sur l'integrabilité des vecteurs gaussiens.Z. Wahsch. verw. Gebiete 68, 1–8.
Acosta, A. de (1985). On the functional form of Lévy's modulus of continuity for Brownian motion.Z. Wahrsch. verw. Gebiete 69, 567–579.
Borell, C. (1975). The Brunn-Minkowski inequality in Gauss space.Invent. Math. 30, 207–216.
Fernique, X. (1970). Integrabilité des vecteurs Gaussiens.C. R. Acad. Sci. Paris 270, 1698–1699.
Seneta, E. (1976).Regularly Varying Functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Goodman, V., Kuelbs, J. Rates of convergence for increments of Brownian motion. J Theor Probab 1, 27–63 (1988). https://doi.org/10.1007/BF01076287
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01076287