Abstract
Let (E, ¦·¦) be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let μ be the convolution of the distribution of a random series inE with independent one-dimensional components and an arbitrary probability measure onE. Under some assumptions about the components and the smoothness of the norm we show that there exists a constant ϰ such that |μ{‖·‖<t}−μ{‖·+r‖<t}|⩽ϰ‖r‖q, whereq depends on the properties of the norm. We specify it in the case ofL α spaces, α>1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bonic, P., and Frampton, J. (1966). Smooth functions on Banach manifolds.J. Math. Mech. 15, 877–898.
Hoffmann-Jørgensen, J., Shepp, L. A., and Dudley, R. M. (1979). On the lower tail of Gaussian seminorms.Ann. Prob. 7, 319–342.
LePage, R., Woodroofe, M., and Zinn, J. (1981). Convergence to a stable distribution via order statistics.Ann. Prob. 9, 624–632.
Lewandowski, M., and Żak, T. (1987). On the density of the distribution ofp-stable seminorms, 0<p<1.Proc. Am. Math. Soc. 100/2, 345–351.
Lindenstrauss, Y., and Tzafriri, L. (1979).Classical Banach Spaces II, Springer-Verlag, Berlin.
Rhee, W. S., and Talagrand, M. (1986). Uniform convexity and the distribution of the norm for a Gaussian measure.Prob. Theory Rel. Fields 71, 59–67.
Ryznar, M. (1988). On density of a stable uniformly convex norm. Preprint.
Żak, T. (1988). A formula for the density of the norm of stable random vectors in Hilbert spaces. to appear inProb. Mathn. Stat.
Żak, T. (1988). On the difference of Gaussian measure of two balls in Hilbert spaces. To appear in Lecture Notes in Math., ProceedingsProbability on Vector Spaces IV, Łańcut 1987.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ryznar, M., Żak, T. The measure of a translated ball in uniformly convex spaces. J Theor Probab 3, 547–562 (1990). https://doi.org/10.1007/BF01046095
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01046095