For branching processes, there are many well-known limit theorems regarding the evolution of the population in the future time. In this dissertation, we investigate the other direction of the evolution, that is, the past of the processes. We pick some individuals at random by simple random sampling without replacement and trace their lines of descent backward in time until they meet. We study the coalescence problem of the discrete-time multi-type Galton-Watson branching process and both the continuoustime single-type and multi-type Bellman-Harris branching processes including the generation number, the death time (in the continuous-time processes) and the type (in the multi-type processes) of the last common ancestor ( also called the most recent common ancestor) of the randomly chosen individuals for the different cases (supercritical, critical, subcritical and explosive).

https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=1011&context=etd

Coalescence in Bellman-Harris and multi-type branching processes

Let \(\mathbf {Z}_{n}=(Z_{n}^{(1)},Z_{n}^{(2)},\cdots ,Z_{n}^{(d)})\) be a d-type (d < ∞) Galton-Watson branching process. For a positive integer k ≥ 2. Pick k individuals at random from the nth generation by simple random sampling without replacement. Trace their lines of descent backward in time till they meet. Let Xn,k be the generation number of the coalescence time of these k individuals of the nth generation. We call the common ancestor of these chosen individuals in the Xn,kth generation their last common ancestor. In this paper, the limit behaviors of the distributions of Xn,k, for any integer k ≥ 2, is studied for the supercritical cases. Also, we investigate the limit distribution of joint distribution of the generation number and the type of the last common ancestor of these randomly chosen individuals and their types in the supercritical case.

Coalescence on Supercritical Multi-type Branching Processes

We present a number of new solutions to an integral equation arising in the limiting theory of Bellman-Harris processes. The argument proceeds via straightforward analysis of Mellin transforms. We also derive a criterion for the analyticity of the Laplace transform of the limiting distribution on Re( u ) ≥ - c for some c > 0.

New limiting distributions for Bellman-Harris processes

Coalescence in the recent past in rapidly growing populations

Let (Z t ) 0 ≤ t 0 is the Malthusian parameter defined by m 0 ∞e- α y G(dy) = 1. In the case Σ ∞ j=1 p j j log j < oo Athreya (1969) has proven analytically that φ(s) is (up to a multiple constant) the unique solution of the above functional equation. In this paper we extend this result by a probabilistic argument assuming only that 1 < m < ∞. We also extend Athreya's (1969) result on the absolute continuity of W to the present case under a mild restriction on G.

On the Supercritical Bellman-Harris Process with Finite Mean

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On the Supercritical Bellman-Harris Process with Finite Mean