Some Exactly Solvable Models of Urn Process Theory
نویسندگان
چکیده
We establish a fundamental isomorphism between discrete-time balanced urn processes and certain ordinary differential systems, which are nonlinear, autonomous, and of a simple monomial form. As a consequence, all balanced urn processes with balls of two colours are proved to be analytically solvable in finite terms. The corresponding generating functions are expressed in terms of certain Abelian integrals over curves of the Fermat type (which are also hypergeometric functions), together with their inverses. A consequence is the unification of the analyses of many classical models, including those related to the coupon collector’s problem, particle transfer (the Ehrenfest model), Friedman’s “adverse campaign” and Pólya’s contagion model, as well as the OK Corral model (a basic case of Lanchester’s theory of conflicts). In each case, it is possible to quantify very precisely the probable composition of the urn at any discrete instant. We study here in detail “semi-sacrificial” urns, for which the following are obtained: a Gaussian limiting distribution with speed of convergence estimates as well as a characterization of the large and extreme large deviation regimes. We also work out explicitly the case of 2-dimensional triangular models, where local limit laws of the stable type are obtained. A few models of dimension three or greater, e.g., “autistic” (generalized Pólya), cyclic chambers (generalized Ehrenfest), generalized coupon-collector, and triangular urns, are also shown to be exactly solvable.
منابع مشابه
Models.
Dynamical urn models, such as the Ehrenfest model, have played an important role in the early days of statistical mechanics. Dynamical many-urn models generalize the former models in two respects: the number of urns is macroscopic, and thermal effects are included. These many-urn models are exactly solvable in the mean-field geometry. They allow analytical investigations of the characteristic f...
متن کاملP´olya Urn Models and Connections to Random Trees: A Review
This paper reviews P´olya urn models and their connection to random trees. Basic results are presented, together with proofs that underly the historical evolution of the accompanying thought process. Extensions and generalizations are given according to chronology: • P´olya-Eggenberger’s urn • Bernard Friedman’s urn • Generalized P´olya urns • Extended urn schemes • Invertible urn schemes ...
متن کاملAnalysis of some exactly solvable diminishing urn models
We study several exactly solvable Pólya-Eggenberger urn models with a diminishing character, namely, balls of a specified color, say x are completely drawn after a finite number of draws. The main quantity of interest here is the number of balls left when balls of color x are completely removed. We consider several diminishing urns studied previously in the literature such as the pills problem,...
متن کاملP´olya-Type Urn Models with Multiple Drawings
We investigate the distribution, mean value, variance and some limiting properties of an urn model of white and red balls under random multiple drawing (either with or without replacement) when the number of white and red balls added follows a schedule that depends on the number of white balls chosen in each drawing.
متن کاملSolution of Urn Models by Generating Functions with Applications to Social, Physical, Biological, and Network Sciences
The primary subject of this thesis is the notion of an urn model and their applications to complex systems. It is demonstrated that several models of social, physical, and biological science are special cases of a large class of urn models that are then exactly solved. These models prescribe 2 or more urns with N balls distributed among them. Two balls are then drawn randomly and then redistrib...
متن کامل