A note on recurrent events

Extract

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 E₁, E₂, …, 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 λ₁ = 1 are less than one (see Bartlett(1)).