Maximal inequalities for centered norms of sums of independent random vectors

Rafał Latała

doi:10.1007/978-3-0348-0490-5_4

paper

Maximal inequalities for centered norms of sums of independent random vectors

arXiv:1501.00698 · doi:10.1007/978-3-0348-0490-5_4

Abstract

Let $X_{1}, X_{2}, \dots, X_{n}$ be independent random variables and $S_{k} = \sum_{i = 1}^{k} X_{i}$ . We show that for any constants $a_{k}$ , \[ \Pr(\max_{1\leq k\leq n}||S_{k}|-a_{k}|>11t)\leq 30 \max_{1\leq k\leq n}\Pr(||S_{k}|-a_{k}|>t). \] We also discuss similar inequalities for sums of Hilbert and Banach space valued random vectors.

9 pages