Existence, uniqueness and constructions for stochastically monotone Q-processes

The key problems in discussing duality and monotonicity for continuous-time Markov chains are to find conditions for existence and uniqueness and then to construct corresponding processes in terms of infinitesimal characteristics, i.e., q-matrices. Such problems are solved in this paper under the assumption that the given q-matrix is conservative. Some general properties of stochastically monotone Q-process ( Q is not necessarily conservative) are also discussed.

Strong ergodicity of monotone transition functions

By revealing close links among strong ergodicity, monotone, and the Feller–Reuter–Riley (FRR) transition functions, we prove that a monotone ergodic transition function is strongly ergodic if and only if it is not FRR. An easy to check criterion for a Feller minimal monotone chain to be strongly ergodic is then obtained. We further prove that a non-minimal ergodic monotone chain is always strongly ergodic. The applications of our results are illustrated using birth-and-death processes and branching processes.

Uniqueness and extinction properties of generalised Markov branching processes

This paper focuses on the basic problems regarding uniqueness and extinction properties for generalised Markov branching processes. The uniqueness criterion is firstly established and a differential–integral equation satisfied by the transition functions of such processes is derived. The extinction probability is then obtained. A closed form is presented for both the mean extinction time and the conditional mean extinction time. It turns out that these important quantities are closely related to the elementary gamma function.

Uniqueness criteria for continuous-time Markov chains with general transition structures

We derive necessary and sufficient conditions for the existence of bounded or summable solutions to systems of linear equations associated with Markov chains. This substantially extends a famous result of G. E. H. Reuter, which provides a convenient means of checking various uniqueness criteria for birth-death processes. Our result allows chains with much more general transition structures to be accommodated. One application is to give a new proof of an important result of M. F. Chen concerning upwardly skip-free processes. We then use our generalization of Reuter's lemma to prove new results for downwardly skip-free chains, such as the Markov branching process and several of its many generalizations. This permits us to establish uniqueness criteria for several models, including the general birth, death, and catastrophe process, extended branching processes, and asymptotic birth-death processes, the latter being neither upwardly skip-free nor downwardly skip-free.

Isomorphic effects of managerial and directoral career paths

The extensive array of interlocking directorate research remains near-exclusively cross-sectional or comparative cross-sectional in nature. While this has been fruitful in identifying persistent structures of inter-organisational relationships evidence of the impact of these structures on organisational performance or activity has been more limited. This should not be surprising because, by their nature, relationships have strong longitudinal and dynamic qualities that are likely to be difficult to isolate through cross-sectional approaches. Clearly, managerial practice is inevitably strongly conditioned by the specific contingencies of the time and the information available through networks of colleagues and advisers (particularly at board level) at the time. But managerial and directoral capabilities and mental sets are also developed over time, particularly through previous experiences in these roles and the formation of long-lasting 'strong' and 'weak' relationships. This paper tests the influence of three longitudinal dimensions of managers and directors' relationships on a set of indicators of financial performance, drawing from a large dataset of detailing historic board membership of UK firms. It finds evidence of isomorphic processes through these channels and establishes that the longitudinal design considerably enhances the detection of performance effects from directorate interlocks. More broadly, the research has implications for the conception of collective action and the constitution of 'community'.