The Living Thing / Notebooks : Branching processes and their statistics

A diverse class of stochastic models that I am mildly obsessed with.

In particular, I am interested in “pure birth” branching processes, where each event leads to certain numbers of offspring with a certain probability. These correspond to certain types of “cluster” and “self excitation” processes.

I will assume linear intensity models except where otherwise indicated.

To learn

Discrete index, discrete state: The Galton-Watson process and friends

There are many standard expositions. Two good ones:

This section got long enough to break out separately. See my notes on long-memory Galton-Watson process.

Continuous index, discrete state: the Hawkes Process

If you have a integer-valued state space, but a continuous time index, and linear intensities, then this is a Hawkes Point Process, the cluster point process. Consider also change of time.

See my masters thesis.

Continuous index, continuous state: The CSBP type of Lévy process

Aldous does a no-nonsense expo on these. Super trendy at the moment: It turns out that growing trees is connected in a deep but purportedly simple way to “glueing together” excursions of random processes, oh, and a bunch of trippy fractals and random trees and stuff.

Lee and Hopcraft (LeHJ08) also found an analogous result for discrete state branching processes.

Inference theory

I’m curious about this, and Lévy process inference in general. It’s interesting because such processes are always incompletely sampled; What’s the best you can do with finitely many samples from a continuous branching process? For the simple case of the Weiner process (as a Lévy process) there is a well-understood estimation theory, with twiddly flourishes on top even. For CSBPs I am not aware of any specific examples except in very specific constrained cases, notably, the discrete state case, which rather defeats the purpose. Can you do more with general nonparametric Lévy measure inference in this continuous case? How much more? Over98 seems to be one of the few refs. Surely the finance folks are onto this?

Discrete index, continuous state

Umm. Is this well-defined? I suppose so. Can’t find any literature references though. It surely has a fancy name. “Marked Galton-Watson Process”? Some kind of compound Poisson, I imagine.

Special issues for multivariate branching processes

If you are looking at cross-excitation between variables then I have some additional matter at contagion processes.


Measure-valued state or something? Are these even branching processes, or did they just seem to be so because I ran into them at the end of a branching process seminar? Can’t recall, must investigate later.

Dynk04, Dynk91 and Ethe00 were recommended to me for this.

Classic data sets


IHSEP is Feng Chen’s software to continuous index, discrete state branching processes.

Spatstat is for spatial point processes.


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