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Two notions of information-singular and strong information-singular random processes were proposed by Berger as processes which are deterministic or negligible in a physically meaningful, information theoretic sense. This paper serves two purposes. First, it shows that strong information-singularity of a random process is equivalent to information-singularity plus a quite different property called recoverability. Secondly, it shows that these properties can be completely characterized in the case where the processes of interest are (jointly) stationary and satisfy a mild integrability condition.