Invariance properties of random vectors and stochastic processes based on the zonoid concept

TL;DR

This paper explores the invariance properties of random vectors and processes using the zonoid concept, focusing on zonoid stationarity and swap invariance, and establishes ergodic theorems for these classes.

Contribution

It introduces and analyzes zonoid-based invariance properties for stochastic processes, extending classical concepts like stationarity and exchangeability.

Findings

Ergodic theorem holds for swap-invariant sequences.

Limits of swap-invariant sequences are characterized.

Zonoid stationarity is a weaker condition than stationarity.

Abstract

Two integrable random vectors $\xi$ and $\xi^*$ in $\mathbb {R}^d$ are said to be zonoid equivalent if, for each $u\in \mathbb {R}^d$, the scalar products $\langle\xi,u\rangle$ and $\langle\xi^*,u\rangle$ have the same first absolute moments. The paper analyses stochastic processes whose finite-dimensional distributions are zonoid equivalent with respect to time shift (zonoid stationarity) and permutation of its components (swap invariance). While the first concept is weaker than the stationarity, the second one is a weakening of the exchangeability property. It is shown that nonetheless the ergodic theorem holds for swap-invariant sequences and the limits are characterised.