  We obtain a Shimorin-Wold-type decomposition for a doubly commuting covariant
representation of a product system of $C^*$-correspondences. This extends a
recent Wold-type decomposition by Jeu and Pinto for a $q$-doubly commuting
isometries. Application to the wandering subspaces of doubly commuting induced
representations is explored, and a version of Mandrekar's Beurling type theorem
is obtained to study doubly commuting invariant subspaces using Fock space
approach due to Popescu.
