Random Evolution of Degradation and Occurrences of Words in Random Sequences of Letters

We illustrate some ideas and results that arise in the analysis of waiting times to occurrences of words in random sampling of letters from an alphabet. The basic mathematical ground is constituted by a special class of discrete-time, irreducible, Markov chains. In particular we consider stochastic comparisons between passage-times into a maximal state for couples of Markov chains belonging to such a class. Our discussion is also related to other elds that, at a first glance, may appear quite far away from the context of words' occurrences. Several other problems of applied probability, in fact, can be analyzed in terms of models that are similar to those of the words context, under suitable conditions. In this respect, we present in particular some models related with ageing, degradation, or reliability-maintenance in discrete-time.