Integrability of Seminorms

TL;DR

This paper investigates the integrability properties of seminorms of stochastic processes, extending known results for Banach space polynomials to a broader class of random elements, and addresses a question from Borell (1979).

Contribution

It introduces a simple technique to establish integrability of seminorms for random elements beyond Banach space polynomial limits, broadening the scope of existing results.

Findings

Proves integrability of seminorms for a large class of stochastic processes.

Extends known polynomial chaos results to more general random elements.

Partially answers a question posed by Borell in 1979.

Abstract

We study integrability and equivalence of L^p-norms of polynomial chaos elements. Relying on known results for Banach space valued polynomials, a simple technique is presented to obtain integrability results for random elements that are not necessarily limits of Banach space valued polynomials. This enables us to prove integrability results for a large class of seminorms of stochastic processes and to answer, partially, a question raised by C. Borell (1979, Seminaire de Probabilites, XIII, 1-3).