892 research outputs found
Dynamics of a birth-death process based on combinatorial innovation
A feature of human creativity is the ability to take a subset of existing
items (e.g. objects, ideas, or techniques) and combine them in various ways to
give rise to new items, which, in turn, fuel further growth. Occasionally, some
of these items may also disappear (extinction). We model this process by a
simple stochastic birth--death model, with non-linear combinatorial terms in
the growth coefficients to capture the propensity of subsets of items to give
rise to new items. In its simplest form, this model involves just two
parameters . This process exhibits a characteristic 'hockey-stick'
behaviour: a long period of relatively little growth followed by a relatively
sudden 'explosive' increase. We provide exact expressions for the mean and
variance of this time to explosion and compare the results with simulations. We
then generalise our results to allow for more general parameter assignments,
and consider possible applications to data involving human productivity and
creativity.Comment: 21 pages, 4 figure
On the convergence of moments in stationary Markov chains
AbstractNecessary and sufficient conditions are given for the convergence of the first moment of functionals of Markov chains
Derivatives of Markov kernels and their Jordan decomposition
We study a particular class of transition kernels that stems from differentiating Markov kernels in the weak sense. Sufficient conditions are established for this type of kernels to admit a Jordan-type decomposition. The decomposition is explicitly constructed. © Heldermann Verlag
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