A space discretization of an integrable long wave-short wave interaction model, called the Yajima-Oikawa system, was proposed in the recent paper arXiv:1509.06996 using the Hirota bilinear method (see also https://link.aps.org/doi/10.1103/PhysRevE.91.062902). In this paper, we propose a Lax-pair representation for the discrete Yajima-Oikawa system as well as its multicomponent generalization also considered in arXiv:1509.06996 and prove that it has an infinite number of conservation laws... We also derive the next higher flow of the discrete Yajima-Oikawa hierarchy, which generalizes a modified version of the Volterra lattice. Relations to two integrable discrete nonlinear Schr\"odinger hierarchies, the Ablowitz-Ladik hierarchy and the Konopelchenko-Chudnovsky hierarchy, are clarified. (read more)