Let X1, X2, · ·· be a stationary sequence of random variables and E1, E2, · ··, EN mutually exclusive events defined on k consecutive X's such that the probabilities of the events have the sum unity. In the sequence Ej1, Ej2, · ·· generated by the X's, the mean waiting time from an event, say Ej1, to a repetition of that event is equal to N (under a mild condition of ergodicity). Applications are given.