A group action on increasing sequences of set-indexed Brownian motions

TL;DR

This paper characterizes set-indexed Brownian motions through their projections on increasing sequences and explores the sequence-independent variation property, providing new insights into their structure and applications.

Contribution

It introduces a novel characterization of set-indexed Brownian motions via projections on increasing sequences and studies the sequence-independent variation property.

Findings

Set-indexed Brownian motion characterized by sequence projections

Sequence-independent variation property analyzed for group stationary processes

Applications demonstrated for the theoretical results

Abstract

We prove that a square-integrable set-indexed stochastic process is a set-indexed Brownian motion if and only if its projection on all the strictly increasing continuous sequences are one-parameter $G$-time-changed Brownian motions. In addition, we study the "sequence-independent variation" property for group stationary-increment stochastic processes in general and for a set-indexed Brownian motion in particular. We present some applications.