Nemirovski's Inequalities Revisited

TL;DR

This paper revisits Nemirovski's inequalities for sums of independent random vectors in Banach spaces, comparing three different methods to derive these moment inequalities.

Contribution

It presents and compares three approaches—deterministic norm inequalities, type and cotype inequalities, and truncation with Bernstein's inequality—to establish Nemirovski's moment inequalities.

Findings

All three approaches are valid and have unique advantages.

The paper clarifies the relationships between different methods.

It enhances understanding of moment inequalities in Banach spaces.

Abstract

An important tool for statistical research are moment inequalities for sums of independent random vectors. Nemirovski and coworkers (1983, 2000) derived one particular type of such inequalities: For certain Banach spaces $(\B,\|\cdot\|)$ there exists a constant $K = K(\B,\|\cdot\|)$ such that for arbitrary independent and centered random vectors $X_1, X_2, ..., X_n \in \B$, their sum $S_n$ satisfies the inequality $ E \|S_n \|^2 \le K \sum_{i=1}^n E \|X_i\|^2$. We present and compare three different approaches to obtain such inequalities: Nemirovski's results are based on deterministic inequalities for norms. Another possible vehicle are type and cotype inequalities, a tool from probability theory on Banach spaces. Finally, we use a truncation argument plus Bernstein's inequality to obtain another version of the moment inequality above. Interestingly, all three approaches have their own merits.