School of Mathematics and Statistics - Research Publications
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ItemEXTINCTION IN LOWER HESSENBERG BRANCHING PROCESSES WITH COUNTABLY MANY TYPES(INST MATHEMATICAL STATISTICS, 2019-10-01)We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes. These are multitype Galton–Watson processes with typeset X={0,1,2,…}, in which individuals of type i may give birth to offspring of type j≤i+1 only. For this class of processes, we study the set S of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector q and whose maximum is the partial extinction probability vector q~. In the case where q~=1, we derive a global extinction criterion which holds under second moment conditions, and when q~<1 we develop necessary and sufficient conditions for q=q~. We also correct a result in the literature on a sequence of finite extinction probability vectors that converge to the infinite vector q~.
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ItemThe time-dependent expected reward and deviation matrix of a finite QBD process(Elsevier, 2019-06-01)Deriving the time-dependent expected reward function associated with a continuous-time Markov chain involves the computation of its transient deviation matrix. In this paper we focus on the special case of a finite quasi-birth-and-death (QBD) process, motivated by the desire to compute the expected revenue lost in a MAP/PH/1/C queue. We use two different approaches in this context. The first is based on the solution of a finite system of matrix difference equations; it provides an expression for the blocks of the expected reward vector, the deviation matrix, and the mean first passage time matrix. The second approach, based on some results in the perturbation theory of Markov chains, leads to a recursive method to compute the full deviation matrix of a finite QBD process. We compare the two approaches using some numerical examples.
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ItemFitting Markovian binary trees using global and individual demographic data(ACADEMIC PRESS INC ELSEVIER SCIENCE, 2019-08)We consider a class of continuous-time branching processes called Markovian binary trees (MBTs), in which the individuals lifetime and reproduction epochs are modelled using a transient Markovian arrival process (TMAP). We develop methods for estimating the parameters of the TMAP by using either age-specific averages of reproduction and mortality rates, or age-specific individual demographic data. Depending on the degree of detail of the available information, we follow a weighted non-linear regression or a maximum likelihood approach. We discuss several criteria to determine the optimal number of states in the underlying TMAP. Our results improve the fit of an existing MBT model for human demography, and provide insights for the future conservation management of the threatened Chatham Island black robin population.
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ItemA pathwise approach to the extinction of branching processes with countably many types(ELSEVIER SCIENCE BV, 2019-03-01)We consider the extinction events of Galton–Watson processes with countably infinitely many types. In particular, we construct truncated and augmented Galton–Watson processes with finite but increasing sets of types. A pathwise approach is then used to show that, under some sufficient conditions, the corresponding sequence of extinction probability vectors converges to the global extinction probability vector of the Galton–Watson process with countably infinitely many types. Besides giving rise to a family of new iterative methods for computing the global extinction probability vector, our approach paves the way to new global extinction criteria for branching processes with countably infinitely many types.