  We consider two nonindependent random fields $\psi$ and $\phi$ defined on a
countable set $Z$. For instance, $Z={\mathbb Z}^d$ or $Z={\mathbb Z}^d\times
I$, where $I$ denotes a finite set of possible "internal degrees of freedom"
such as spin. We prove that, if the cumulants of both $\psi$ and $\phi$ are
$\ell_1$-clustering up to order $2 n$, then all joint cumulants between $\psi$
and $\phi$ are $\ell_2$-summable up to order $n$, in the precise sense
described in the text. We also provide explicit estimates in terms of the
related $\ell_1$-clustering norms, and derive a weighted $\ell_2$-summation
property of the joint cumulants if the fields are merely $\ell_2$-clustering.
One immediate application of the results is given by a stochastic process
$\psi_t(x)$ whose state is $\ell_1$-clustering at any time $t$: then the above
estimates can be applied with $\psi=\psi_t$ and $\phi=\psi_0$ and we obtain
uniform in $t$ estimates for the summability of time-correlations of the field.
The above clustering assumption is obviously satisfied by any
$\ell_1$-clustering stationary state of the process, and our original
motivation for the control of the summability of time-correlations comes from a
quest for a rigorous control of the Green-Kubo correlation function in such a
system. A key role in the proof is played by the properties of non-Gaussian
Wick polynomials and their connection to cumulants.
