Orientable embedding of Cayley graphs

Thus any (standard) regenerative event can be represented by a Markov process on a continuous state space. It follows, for example, that the fact that poo(t) is almost everywhere differentiate is a consequence of the regenerative property of the state 0, but the deeper result that poo(t) is everywhere differentiate requires also the discrete nature of the state space. It is possible to extend the whole theory to take in properties of several states simultaneously, by considering systems of regenerative events. In particular, we can examine the transition probabilities pij(t) (i^j) of a Markov chain. The theory may also be applied to certain Markov processes with continuous state space, and so, via the method of supplementary variables, to some non-Markovian processes. It is hoped to publish elsewhere a detailed account of the theory summarised here, and of its various applications. I am deeply grateful to Professor D. G. Kendall for much helpful discussion, and also to the Department of Scientific and Industrial Research for financial support.