We introduce a new kind of random stationary, scalar beam named Laguerre-cosine-Gaussian correlated Schell-model (LcGCSM) beam whose spectral degree of coherence (SDOC) is a Gaussian function modulated by both non-conventional Laguerre-and cosine-type factors. The analytical expressions for the cross-spectral density of a LcGCSM beam propagating through a paraxial ABCD optical system are derived. The intensity distributions of such beams focused through a thin lens are illustrated numerically. It is found that a LcGCSM beam exhibits selfsplitting and self-shaping properties during propagation, evolving into multiple optical cages near the focal region. By directly modulating the spatial structure of the SDOC in the source plane, the characteristics of the optical cages derived from the proposed LcGCSM beams can be adjusted flexibly. Moreover, we demonstrate that these nontrivial focusing properties of a LcGCSM beam can be well elucidated with the combination effect of individual merits associated with each non-conventional SDOC component of the entire SDOC. Therefore, our results provide a new route for generating and controlling multiple optical cages, and will be useful for trapping multiple particles, guiding multiple atoms and optical communications.