We consider a discrete-time semi-Markov process, with a finite state space. Taking a censored history, we obtain empirical estimators for the semi-Markov kernel, semi-Markov transition function, reliability and availability. We study the strong consistency and the asymptotic normality for each estimator.

Let a discrete time semi-Markov process {Z _γ;γ ∈ ℕ} with finite state space an alphabet Ω. Defining the process {U _γ; γ ∈ ℕ} to be the backward recurrence time of the process {Z _γ; γ ∈ ℕ}, we study the Markov process {(Z _γ, U _γ); γ ∈ ℕ}. We give its transition probabilities of first and higher order, the limiting distribution, and the stationary distribution. Using this Markov process we construct a k-dimensional process and we study its basic properties. As an application we consider a finite set of words W = {w ₁, w ₂,…, w _ν} of equal length k which are produced under the semi-Markovian hypothesis and we focus on the waiting time for the first word occurrence from the set W. The corresponding probability distribution, the generating function, as well as the mean waiting time and variance are obtained.