IntroductionLet {P t ; t ^ 0} be a uniformly continuous Markov semi-group, that is, the semigroup formed from the transition probabilities of a discrete state, continuous time Markov process with bounded infinitesimal generator. Then in an earlier paper, [1], it was shown that, for small enough time values t, the operator P t can only arise in the unique uniformly continuous Markov semi-group {P t }.In this paper we impose the additional assumption that in fact {P t } has a finite number of states, and consider the more general question of in how many Markov semi-groups the operator P t , for a fixed t, can arise. Specifically, if we define the function L(t) to be the number of distinct Markov semi-groups coinciding with our given semi-group at time t, we are interested in the form of the function L. The key to our results on this question is the detailed spectral decomposition available for finite matrices. In §2, after stating a general result on matrix exponentials, a theorem is proved on the structure of the function L.In Section 3 we consider the question of delimiting the set {t; L(f) = 1} more precisely. This involves the problem of putting an upper bound on the norm of the infinitesimal generator of the semi-group from knowledge of P t for a single fixed t > 0: one method of doing this is derived, and then this bound is applied to determine an interval on which L{t) = 1.It is known that, for two-state Markov semi-groups, the function L(t) = 1. The three-state case is, on the other hand, non-trivial, but we show in the final section that it does involve certain simplifications, both for the structure of the logarithm function, and in the related imbedding problem.

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An important problem in road safety is to estimate the rate of involvement of drivers in accidents per unit of exposure, or road usage. Typically, good data on numbers of drivers involved are available for the numerator of this ratio, but data on road usage for the denominator are often not available. To circumvent this difficulty, techniques of induced exposure were developed in the 1960s, to derive estimates of rates of driver involvement solely from data on numbers of drivers. None of the existing variants of the induced exposure technique seems entirely satisfactory, however. This paper derives a new version of the induced exposure approach and illustrates its application to accident data for Scotland over the period 1986-90.