Higher moments of Banach space valued random variables

TL;DR

This paper explores the definition and properties of higher moments of Banach space valued random variables using tensor products and various types of integrals, with implications for probability metrics and the contraction method.

Contribution

It introduces a comprehensive framework for defining and analyzing higher moments in Banach spaces, comparing projective and injective tensor products, and examining their properties across different spaces.

Findings

Injective moments do not always determine projective moments in general Banach spaces.

The approximation property of Banach spaces ensures equivalence of certain moments.

Results are specialized for Hilbert spaces, $C(K)$, and $D[0,1]$, including measurability issues.

Abstract

We define the $k$:th moment of a Banach space valued random variable as the expectation of its $k$:th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. We study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals. One of the problems studied is whether two random variables with the same injective moments (of a given order) necessarily have the same projective moments; this is of interest in applications. We show that this holds if the Banach space has the approximation property, but not in general. Several sections are devoted to results in special Banach spaces, including Hilbert spaces, $C(K)$ and $D[0,1]$. The latter space is non-separable, which complicates the arguments, and we prove various preliminary results on e.g. measurability in $D[0,1]$ that we need. One of the main motivations of this paper is the application to Zolotarev metrics and their use in the contraction method. This is sketched in an appendix.