A note on recurrent events

Consider a Markoff chain (by which is meant a stochastic process defined for discrete values of both variable and parameter, and whose probability dependence does not extend to more than a unit interval) with completely specified transition probability matrix P = ( p ij ), where p ij is the conditional probability that the ( r + 1)th observation belongs to the state E i , given that the rth observation belongs to the state E i for r ≥ I. We assume that there are a states E 1 , E 2 , …, E a , where a is finite. We also assume an ergodic property for the stochastic process, that is, we consider non-periodic chains for which all the possible initial states remain permanently available. This defines what has been called the positively regular case. Also under the above assumption the moduli of all the latent roots λ r of the transition probability matrix P = ( p ij ) other than the first simple latent root λ 1 = 1 are less than one (see Bartlett(1)).