Let, for each t∈T, ψ(t, ۔) be a random measure on the
Borel σ-algebra in ℝd such that Eψ(t, ℝd)k < ∞ for all k and let $\widehat{\psi}$(t, ۔) be
its characteristic function. We call the function
$\widehat{\psi}$
(t1,…, tl ; z1,…, zl) = ${\sf E}\prod^l_{j=1}\widehat{\psi}(t_j, z_j)$ of arguments l∈ ℕ, t1, t2… ∈T, z1, z2∈ ℝd the covaristic of the measure-valued random function (MVRF)
ψ(۔, ۔). A general limit theorem for MVRF's in
terms of covaristics is proved and applied to functions of the
kind
ψn(t, B) = µ{x : ξn(t, x) ∈B}, where μ is a
nonrandom finite measure and, for each n, ξn is a
time-dependent random field.