14. The generating-function argument used to prove Theorem 10.12 has a powerful application to the general theory of recurrent events. Let be an event

14. The generating-function argument used to prove Theorem 10.12 has a powerful application to the general theory of ‘recurrent events’. Let η be an event which may or may not happen at each of the time points 0, 1, 2, . . . (η may be the visit of a random walk to its starting point, or a visit to the dentist, or a car accident outside the department). We suppose that η occurs at time 0. Suppose further that the intervals between successive occurrences of η are independent, identically distributed random variables X1, X2, . . . , each having mass function P(X = k) = fk for k = 1, 2, . . . , so that η occurs at the times 0, X1, X1 + X2, X1 + X2 + X3, . . . . There may exist a time after which η never occurs. That is to say, there may be an Xi which takes the value ∞, and we allow for this by requiring only that f = f1 + f2 + · · · satisfies f ≤ 1, and we set P(X = ∞) = 1 − f.