Maximal inequalities for centered norms of sums of independent random vectors

TL;DR

This paper establishes maximal inequalities for centered norms of sums of independent random vectors, providing bounds that relate the maximum deviation to individual deviations, with extensions to Hilbert and Banach space valued vectors.

Contribution

It introduces a new maximal inequality for centered norms of sums of independent vectors and extends the results to Hilbert and Banach space valued vectors.

Findings

Maximal inequality bounds the probability of large deviations of the maximum centered norm.

Inequalities are applicable to sums of vectors in Hilbert and Banach spaces.

Provides explicit constants for the inequalities.

Abstract

Let $X_1,X_2,\ldots,X_n$ 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.