We introduce a new class of vector array beams named radially polarized Gaussian Schell-model array (RPGSMA) beams for producing far fields with controllable optical lattice average intensity patterns, and derive the realizability conditions for a RPGSMA source and the beam condition for radiation produced by this source based on the unified theory of coherence and polarization, Moreover, the statistical properties of such beam propagating in free space are investigated in detail. It is found that, the statistical properties of the RPGSMA beams on propagation are closely related to source parameters of the beam. with an appropriate choice of the source parameters, the average intensity will produce far fields with controllable optical lattice average intensity patterns, and the degree of coherence (DOC) from the lattice distribution on the original plane evolves into a Gaussian profile in the far field, Furthermore, the degree of polarization (DOP) appears a periodical grid-like distribution on propagation. These results may be useful in optical trapping and free-space optical communications.