Sample Path Properties of the Average Generation of a Bellman-Harris Process

TL;DR

This paper proves strong convergence properties for the average generation in super-critical Bellman-Harris processes, including a two-type model, supporting DNA-based cell lineage inference methods.

Contribution

It establishes the long-term accuracy of an estimation method for average cell generations in complex Bellman-Harris processes, extending to multi-type cellular development.

Findings

Strong convergence results for average generation in super-critical Bellman-Harris processes.

Extension of results to two-type processes with asymmetric offspring.

Validation of the estimation method’s long-term accuracy for cellular lineage analysis.

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.