It is shown that, for any n x n stochastic matrix A=(aij), #(6ij) which can be imbedded in a continuous time Markov chain, there exist distinct indices i,j such that for all k aik = 0 implies a~k = 0, and likewise distinct indices i',j' such that, for all k, aki,=-O implies akj,=O.
The present proof of this does not use Kingman and Williams' characterization of the patterns of zero entries which can occur in imbeddable stochastic matrices.
In the analogous doubly stochastic situation the same result holds, even with "implies" replaced by "if and only if". The main result is that the set Y of imbeddable stochastic (or doubly stochastic) matrices is a Lipschitz manifold with boundary. For any Markov chain leading to a matrix on the boundary of ~ the associated intensity matrix in the Kolmogorov differential equation has, at almost every time, at least one zero entry.
B. Fuglede
with boundary ~" (Theorems 1.13 and 1.16 in the doubly stochastic case). The tangent space J-to Off at some A~0ff thus exists for almost every Ar (with respect to surface measure), and we find that @ (as a vector space) is tangent to each of the convex cones AS and .@A. We write ~=IR+(~-I), denoting the convex semigroup of all n x n stochastic (resp. doubly stochastic) matrices, and I the identity matrix. Our analysis also shows that, for any A e ~,~, every Markov chain (s, t)~ P (s, t) leading from I to P (0, to)= A is confined entirely to ~,~, and the associated intensity matrices P(0, t)-~?P(O, t)/Ot=-(~P(t, to)/at) P(t, to) -1, defined a.e. for re[0, to] (when using Goodman's intrinsic time [9]), belong to the topological boundary ~ of .@ (Theorem 1.14).
The doubly stochastic case is studied in Sect. 1, and the singly stochastic case -which is very similar -is discussed in Sect. 2. Relations to known results in the literature are described in Sect. 3.
I wish to thank Arne Brondsted, Soren Johansen, and Paul Ressel for valuable information and discussions.