Universal generalized functionals and finitely absolutely continuous measures on Banach spaces

TL;DR

This paper explores the convergence of random process functionals to generalized functionals in Banach spaces, emphasizing uniform finite absolute continuity with respect to Wiener measure, unifying various examples under this framework.

Contribution

It introduces a unified approach to convergence of measures in Banach spaces based on uniform finite absolute continuity, connecting diverse examples of process functionals.

Findings

Limit functionals are finitely absolutely continuous w.r.t. Wiener measure.

Convergence characterized by uniform finite absolute continuity.

Unified framework for various examples of process functional convergence.

Abstract

In this paper we collect several examples of convergence of functions of random processes to generalized functionals of those processes. We remark that the limit is always finitely absolutely continuous with respect to Wiener measure. We try to unify those examples in terms of convergence of probability measures in Banach spaces. The key notion is the condition of uniform finite absolute continuity.