  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.
