We prove that the probability substitution matrices obtained from a continuous-time Markov chain form a multiplicatively closed set if and only if the rate matrices associated to the chain form a linear space spanning a Lie algebra. The key original contribution we make is to overcome an obstruction, due to the presence of inequalities that are unavoidable in the probabilistic application, that prevents free manipulation of terms in the Baker-Campbell-Haursdorff formula.
- Pub Date:
- April 2017
- Mathematics - Group Theory;
- Mathematics - Statistics Theory;
- Quantitative Biology - Populations and Evolution;
- Quantitative Biology - Quantitative Methods
- v2: 6 pages. Minimality condition included in Property 0 to close gap in the proof of main result. To appear in the ANZIAM Journal