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This paper shows that Harris recurrent Markov chains and processes can be characterized as the class of Markov chains and processes for which there exists a random time <i>T</i> at which the distribution of the chain or process does not depend on its initial condition. In particular, no independence assumptions concerning the post-<i>T</i> process or <i>T</i> play a role in the characterization. Since Harris chains and processes are known to contain infinite sequences of regeneration times exhibiting various independence properties, it follows that the existence of this single <i>T</i> implies the existence of infinitely many times at which regeneration occurs.