First Passage and Recurrence Distributions

See Feller [4, Chap. 15] for the relevant theory. Let da be defined as the probability that the state, initially supposed to be i, takes on the value j at least once before returning to i. The quantities da turn out to be very useful. In §2 we derive some identities to be used in the sequel. In §3 we consider the distribution of the recurrence time Nkk under the assumptions (1.2) and (1.3), for "rare" states—i.e., states for which tt* is small. Since (assuming that there are infinitely many states) no matter how the states are numbered, we must have it*—»0 as k—>&>, we can speak of the distribution of Nkk for large k. It is shown that