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

Abstract

Two integrable random vectors ξ and ξ* in IRd are said to be zonoid equivalent if, for each u∈IRd, the scalar products 〈ξ,u〉 and 〈ξ*,u〉 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 time moments (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 characterized.