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Sample path properties of the average generation of a Bellman–Harris process

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Abstract

Motivated by a recently proposed design for a DNA coded randomised algorithm that enables inference of the average generation of a collection of cells descendent from a common progenitor, here we establish strong convergence properties for the average generation of a super-critical Bellman–Harris process. We further extend those results to a two-type Bellman–Harris process where one type can give rise to the other, but not vice versa. These results further affirm the estimation method’s potential utility by establishing its long run accuracy on individual sample-paths, and significantly expanding its remit to encompass cellular development that gives rise to differentiated offspring with distinct population dynamics.

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Acknowledgements

This work was supported by Science Foundation Ireland Grant 12 IP 1263.

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  1. Hamilton Institute, Maynooth University, Co. Kildare, Ireland

    Gianfelice Meli & Ken R. Duffy

  2. The Walter and Eliza Hall Institute of Medical Research, The University of Melbourne, Parkville, Australia

    Tom S. Weber

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  1. Gianfelice Meli
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Meli, G., Weber, T.S. & Duffy, K.R. Sample path properties of the average generation of a Bellman–Harris process. J. Math. Biol. 79, 673–704 (2019). https://doi.org/10.1007/s00285-019-01373-0

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