Poisson processes (possibly nonhomogeneous) are constructed in the function spaces D q ≡D([0, 1] q , R) and Dq q x ⋯ x D q in order to approximate superpositions of uniformly sparse point processes and partial sums of infinitesimal integer-valued nonnegative random variables. Bounds for the Prohorov distance are computed, where the Prohorov distance is defined on the space of all probability measures on D q, with the Skorohod metric being used on D q. These bounds yield functional central limit theorems (invariance principles) and rates of convergence for functional central limit theorems involving convergence to the Poisson process. In this regard, this paper is an extension of Section 6 of Dudley [4].