Galton–Watson

Galton–Watson
Public domain · source
Name	Galton–Watson process
Type	Stochastic process
Field	Probability theory
Introduced	19th century
Founder	Francis Galton; Henry William Watson

Galton–Watson

Introduction

The Galton–Watson process originated in work by Francis Galton and Henry William Watson and influenced research by Andrey Kolmogorov, Andrei N. Kolmogorov, William Feller, Joseph L. Doob, Kiyoshi Itô, M. S. Bartlett, A. M. Yaglom, Ronald A. Fisher, Harald Cramér, Otto Hölder, Pál Erdős, Paul Erdős, Alfréd Rényi, John von Neumann, Norbert Wiener, Émile Borel, George Pólya, William Rowan Hamilton, Arthur Cayley, James Joseph Sylvester, Augustin-Louis Cauchy, Srinivasa Ramanujan, Stanislaw Ulam, John Maynard Keynes, Harold Hotelling, Abraham Wald, Jerzy Neyman, Egmont Colerus, Karl Pearson, G. H. Hardy, David Hilbert, André Weil, Alexander Grothendieck, Paul Lévy, Kurt Gödel, Alonzo Church, Alan Turing, Claude Shannon, Norbert Frisch, Hermann Weyl, Eugene Wigner, John Nash, Emmy Noether, Sophus Lie, Felix Hausdorff, Léon Walras influenced probabilistic thinking that contextualizes branching processes and drew attention from scholars at institutions such as University of Cambridge, Trinity College, Cambridge, University of Oxford, Imperial College London, University of Chicago, Princeton University, Harvard University, University of Paris, École Normale Supérieure, University of Göttingen, University of Vienna, ETH Zurich, University of Warsaw, Sorbonne University, Columbia University, Massachusetts Institute of Technology, Stanford University, California Institute of Technology, University of California, Berkeley, University of Edinburgh, University of Manchester, University of Melbourne, University of Tokyo, and National University of Singapore.

Definition and model

The classical discrete-time branching framework considers a root ancestor whose offspring count follows a fixed probability law; early expositors included Francis Galton, Henry William Watson, Andrey Kolmogorov, William Feller, Joseph L. Doob, Kiyoshi Itô, and M. S. Bartlett who formalized generating-function techniques that mirror methods used by Paul Erdős, Alfréd Rényi, G. H. Hardy, Srinivasa Ramanujan, David Hilbert, John von Neumann, Claude Shannon, Alan Turing, Norbert Wiener, Émile Borel, and George Pólya. The model is specified by an initial population size and an offspring distribution often studied via contributions from Pál Erdős, Paul Lévy, Ronald A. Fisher, Harald Cramér, André Weil, Alexander Grothendieck, and Kurt Gödel in analogous probabilistic contexts. Mathematically oriented treatments appear in classic texts by William Feller, Joseph L. Doob, Andrey Kolmogorov, M. S. Bartlett, Kiyoshi Itô, John von Neumann, and Norbert Wiener.

Extinction probability and generating functions

Extinction questions were central to analyses by Francis Galton and Henry William Watson, later advanced by Andrey Kolmogorov, William Feller, Joseph L. Doob, Kiyoshi Itô, Pál Erdős, and A. M. Yaglom using probability-generating functions similar to generating function methods in work by G. H. Hardy, John von Neumann, Claude Shannon, Alan Turing, Émile Borel, George Pólya, David Hilbert, Paul Lévy, Harald Cramér, and Ronald A. Fisher. Criticality classification (subcritical, critical, supercritical) appears in treatises by Andrey Kolmogorov, William Feller, Joseph L. Doob, Kiyoshi Itô, M. S. Bartlett, and A. M. Yaglom and connects with limit theorems discussed by Paul Erdős, Alfréd Rényi, John Nash, Emmy Noether, Felix Hausdorff, Hermann Weyl, and Eugene Wigner. Analytical techniques draw on fixed-point methods and complex-analytic approaches used by David Hilbert, Émile Borel, John von Neumann, G. H. Hardy, and André Weil.

Variants and generalizations

Multiple-lineage and multitype extensions were developed with input from M. S. Bartlett, Kiyoshi Itô, A. M. Yaglom, William Feller, Joseph L. Doob, Andrey Kolmogorov, Paul Erdős, Alfréd Rényi, Stanislaw Ulam, John von Neumann, Norbert Wiener, Claude Shannon, Alan Turing, Jerzy Neyman, Abraham Wald, Ronald A. Fisher, and Harald Cramér. Continuous-time branching processes and age-dependent models relate to work by Kiyoshi Itô, Andrey Kolmogorov, William Feller, Joseph L. Doob, Paul Lévy, André Weil, Alexander Grothendieck, and Pál Erdős. Spatial branching, branching random walks, and branching Brownian motion have been linked to research at University of Cambridge, Princeton University, Harvard University, Stanford University, ETH Zurich, and University of Paris and analyzed by John Nash, Stanislaw Ulam, Eugene Wigner, Hermann Weyl, Claude Shannon, Alan Turing, Norbert Wiener, and Kiyoshi Itô.

Applications and historical significance

Originally motivated by concerns raised by Francis Galton about hereditary surnames, the framework informed studies in demography and genetics pursued by Ronald A. Fisher, Harald Cramér, Karl Pearson, John Maynard Keynes, Jerzy Neyman, Abraham Wald, M. S. Bartlett, and A. M. Yaglom. Its mathematical structure influenced probability theory developed by Andrey Kolmogorov, William Feller, Joseph L. Doob, Kiyoshi Itô, Paul Lévy, and G. H. Hardy and found applications in research at University of Cambridge, University of Oxford, University of Chicago, Princeton University, Harvard University, Imperial College London, Stanford University, College de France, École Normale Supérieure, ETH Zurich, University of Warsaw, University of Göttingen, and University of Vienna. Later interdisciplinary uptake occurred in ecology and epidemiology studied by John Maynard Keynes, Ronald A. Fisher, Jerzy Neyman, Harald Cramér, Paul Erdős, Alfréd Rényi, Stanislaw Ulam, Claude Shannon, Alan Turing, and Norbert Wiener, and the model informs modern work in computational biology, network science, and information theory at Massachusetts Institute of Technology, California Institute of Technology, University of California, Berkeley, Columbia University, National University of Singapore, University of Tokyo, University of Melbourne, University of Manchester, University of Edinburgh, and Sorbonne University.

Category:Probability theory