The measure of a translated ball in uniformly convex spaces (Q923464)

Let E be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let \(\mu\) be the convolution of the distribution of a random series in E with independent, one-dimensional components and an arbitrary probability measure on E. Under some assumptions on the components and the smoothness of the norm, the authors show that there exists a constant c such that for every \(r\in E\) and \(t\geq 0,\) \[ | \mu (B_ t)-\mu (B_ t+r)| \leq c\| r\|^ q, \] where \(B_ t\) is an open ball with radius t and q depends on the properties of the norm. This is an extension of an earlier result of the second author for symmetric Gaussian measures on separable Hilbert spaces [Probab. Math. Stat. 10, No.2, 257-270 (1989)]. As an application of the above result the authors investigate the obtained estimate for symmetric Gaussian and stable measures on E.