In this talk we will consider a rubric, laid out by Hardy and Harris, under which many earlier formulations of distinguished path analysis (or "spine techniques") for branching processes are unified.
It has been known for some time that there is a connection between single-particle martingales and certain additive martingales for the corresponding branching processes. This connection is made explicit under the Hardy/Harris framework, where each is seen to be the projection of a single, more general martingale onto different sub sigma-algebras. We will also see a nice, intuitive formulation of the martingale change of measure that results in the typical alterations along the spine: namely a change in the drift, a change in the offspring distribution ("size biasing"), and a change in the reproductive rate.
The setting and machinery laid out by Hardy and Harris are quite elegant, and create a very natural vantage from which to approach distinguished path analysis. The power of their framework will be demonstrated through consideration of example applications.