A semi-Markov process X(t) may be viewed as constructed from a two-dimensional process J(,n),W(,n) with (UNFORMATTED TABLE FOLLOWS); X(t) = J(,N(t)) (1a)(TABLE ENDS);where;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);Here J(,n) is a discrete-state Markov chain describing the succession of visited states, and W(,n) is a process describing the succession of times spent in these states, with the W(,n) independent conditionally on realization of J(,n) ;The limiting state distribution of a semi-Markov process is attrac- tively structured in terms of the limiting distribution of J(,n) and the expectations of the residence times W(,n);In this dissertation certain generalizations of semi-Markov proc- esses (here called "Delay processes") are constructed as in (1), but with J(,n) not necessarily a Markov chain and the W(,n) not necessarily conditionally independent, by expanding on a certain previously suggested elementary ergodic argument;The above-mentioned attractive limiting structure can essen- tially be maintained for Delay processes if attention is focused on "essentially finite" state spaces in the sense of requiring a finite expected number of distinct visited states, and if distributional convergence is demanded in only the Cesaro sense;Two examples of Delay processes are given, for the first of which J(,n) is not Markov of any order, with a state space containing only two points; for the second example, which treats an "inventory position process" in the "lost sales case," J(,n) is not an ergodic Markov chain and the W(,n) are not conditionally independent;Finally, a sense is also given in which a Delay process is "close enough" to a semi-Markov process for ordinary, as opposed to Cesaro, convergence to apply.