\(\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}}\)
Stochastic processes
indexed by time whose state is a discrete (possibly only countable) measure.
Popular in, for example, mathematical models of alleles in
biological evolution.
Population genetics keywords, in approximate order of generality and chronology:
Fisher-Wright diffusion, Moran process, Viot-Fleming process.
Obviously there are many other processes matching this broad description.
For one example, any element-wise-positive vector-valued stochastic process can be
made into a discrete-measure-valued process by normalising the state vector to
sum to 1.
For another, count time series are realisations of the measures of such as process of these.
Chinese-restaurant processes as processes in time presumably fit here, although
AFAICT the use of these processes in the literature is usually not the
time-evolving construction, but rather the infinite-time limit of such a process,
which is confusing nomenclature.
If the process does not take values in discrete measures, that would be a
different notebook, which does not at the moment exist;
The Dirichlet process, despite its name is not usually sampled as a time process
and in any case has boring dynamics. However, it does I believe have
time-indexed extensions (Griffin and Steel 2006; Dunson and Park 2008)
References
Asselah, Amine, Pablo A. Ferrari, and Pablo Groisman. 2011.
“Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces.” Journal of Applied Probability 48 (2): 322–32.
https://doi.org/10.1239/jap/1308662630.
Donnelly, Peter, and Thomas G. Kurtz. 1996.
“A Countable Representation of the Fleming-Viot Measure-Valued Diffusion.” The Annals of Probability 24 (2): 698–742.
http://www.jstor.org/stable/2244946.
Dunson, David B, and Ju-Hyun Park. 2008.
“Kernel Stick-Breaking Processes.” Biometrika 95 (2): 307–23.
https://doi.org/10.1093/biomet/asn012.
Ethier, S. N., and R. C. Griffiths. 1993.
“The Transition Function of a Fleming-Viot Process.” The Annals of Probability 21 (3): 1571–90.
http://www.jstor.org/stable/2244588.
Ethier, S. N., and Thomas G. Kurtz. 1993.
“Fleming–Viot Processes in Population Genetics.” SIAM Journal on Control and Optimization 31 (2): 345–86.
https://doi.org/10.1137/0331019.
Fleming, Wendell H, and Michel Viot. 1979.
“Some Measure-Valued Markov Processes in Population Genetics Theory.” Indiana University Mathematics Journal 28 (5): 817–43.
http://www.jstor.org/stable/24892583.
Griffin, J. E., and M. F. J. Steel. 2006.
“Order-Based Dependent Dirichlet Processes.” Journal of the American Statistical Association 101 (473): 179–94.
https://doi.org/10.1198/016214505000000727.
Konno, N., and T. Shiga. 1988.
“Stochastic Partial Differential Equations for Some Measure-Valued Diffusions.” Probability Theory and Related Fields 79 (2): 201–25.
https://doi.org/10.1007/BF00320919.
Marzen, S. E., and J. P. Crutchfield. 2020.
“Inference, Prediction, and Entropy-Rate Estimation of Continuous-Time, Discrete-Event Processes.” May 7, 2020.
http://arxiv.org/abs/2005.03750.
Moran, P. a. P. 1958.
“Random Processes in Genetics.” Mathematical Proceedings of the Cambridge Philosophical Society 54 (1): 60–71.
https://doi.org/10.1017/S0305004100033193.
Nowak, Martin A. 2006. Evolutionary Dynamics: Exploring the Equations of Life. Belknap Press of Harvard University Press.
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