scholarly journals Gaussian process approximations for multicolor Pólya urn models

2021 ◽  
Vol 58 (1) ◽  
pp. 274-286
Author(s):  
Konstantin Borovkov
Keyword(s):  
Gaussian Process ◽  
Tissue Growth ◽  
Maximum Deviation ◽  
Borel Sets ◽  
Pólya Urn ◽  
Large Numbers ◽  
Long Time ◽  
Empirical And Quantile Processes ◽  
Polya Urn ◽  
Normal Law

AbstractMotivated by mathematical tissue growth modelling, we consider the problem of approximating the dynamics of multicolor Pólya urn processes that start with large numbers of balls of different colors and run for a long time. Using strong approximation theorems for empirical and quantile processes, we establish Gaussian process approximations for the Pólya urn processes. The approximating processes are sums of a multivariate Brownian motion process and an independent linear drift with a random Gaussian coefficient. The dominating term between the two depends on the ratio of the number of time steps n to the initial number of balls N in the urn. We also establish an upper bound of the form $c(n^{-1/2}+N^{-1/2})$ for the maximum deviation over the class of convex Borel sets of the step-n urn composition distribution from the approximating normal law.

Order statistics and the Pólya urn process

1977 ◽  
Vol 9 (2) ◽  
pp. 205-206
Author(s):  
Herbert Robbins ◽  
John Whitehead
Keyword(s):  
Order Statistics ◽  
Pólya Urn ◽  
Polya Urn

GLOBAL FLUCTUATIONS FOR LINEAR STATISTICS OF β-JACOBI ENSEMBLES

2012 ◽  
Vol 01 (04) ◽  
pp. 1250013 ◽  
Author(s):  
IOANA DUMITRIU ◽  
ELLIOT PAQUETTE
Keyword(s):  
Gaussian Process ◽  
Covariance Matrix ◽  
Law Of Large Numbers ◽  
Test Functions ◽  
Linear Statistics ◽  
Jacobi Ensemble ◽  
Large Numbers ◽  
The Mean ◽  
Shifted Chebyshev Polynomial ◽  
Global Fluctuations

We study the global fluctuations for linear statistics of the form [Formula: see text] as n → ∞, for C1 functions f, and λ1, …, λn being the eigenvalues of a (general) β-Jacobi ensemble. The fluctuation from the mean [Formula: see text] turns out to be given asymptotically by a Gaussian process. We compute the covariance matrix for the process and show that it is diagonalized by a shifted Chebyshev polynomial basis; in addition, we analyze the deviation from the predicted mean for polynomial test functions, and we obtain a law of large numbers.

SYNCHRONIZATION IN RANDOMLY DRIVEN SYSTEMS

1995 ◽  
Vol 09 (16) ◽  
pp. 985-988 ◽  
Author(s):  
A.M. JAYANNAVAR
Keyword(s):  
Simple Model ◽  
Law Of Large Numbers ◽  
Driven Systems ◽  
Large Numbers ◽  
Long Time ◽  
The Law

We have solved analytically a simple model of evolution of particles driven by identical noise. We show that the trajectories of all particles collapse into a single trajectory at long time. This synchronization also leads to violation of the law of large numbers.

scholarly journals Administrative Background and the Process of Migrant Integration in Hungary

2018 ◽  
Vol 18 (3) ◽  
pp. 421-446
Author(s):  
István Temesi
Keyword(s):  
Public Administration ◽  
Geographical Location ◽  
Policy Makers ◽  
Legal Rules ◽  
Eu Member States ◽  
Mass Migration ◽  
Large Numbers ◽  
Long Time ◽  
Political Decisions ◽  
The Eu

Some EU member states have been migrant destinations for a long time, while others have lost a considerable part of their population since their accession to the EU. Hungary belongs to the latter. Large numbers of immigrants have not been arriving here since the end of the war in former Yugoslavia. However, in 2015 Hungary was suddenly strongly affected by mass migration, mainly because of the country’s geographical location. Mass migration has strongly influenced politics as the decision-maker and public administration as the executor of political decisions. Both the decisions and the policy-makers have been strongly criticised for taking a different approach to the situation compared with many other European countries. The Hungarian government’s priority was to reduce or stop mass migration and it used political, legal, and physical instruments selected for this purpose. This study does not aim to judge whether they are right or wrong. Hungarian public administration has had to adapt to the situation and it has done so by way of implementing new and modified legal rules. However, due to the political decisions described above, it has developed and changed at the same time.

scholarly journals Predictive Constructions Based on Measure-Valued Pólya Urn Processes

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2845
Author(s):  
Sandra Fortini ◽  
Sonia Petrone ◽  
Hristo Sariev
Keyword(s):  
Polish Space ◽  
Asymptotic Properties ◽  
Transition Kernel ◽  
Urn Model ◽  
Additive Structure ◽  
Predictive Distributions ◽  
Pólya Urn ◽  
Random Weights ◽  
Random Transition ◽  
Polya Urn

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP (μn)n≥0 on a Polish space X, the normalized sequence (μn/μn(X))n≥0 agrees with the marginal predictive distributions of some random process (Xn)n≥1. Moreover, μn=μn−1+RXn, n≥1, where x↦Rx is a random transition kernel on X; thus, if μn−1 represents the contents of an urn, then Xn denotes the color of the ball drawn with distribution μn−1/μn−1(X) and RXn—the subsequent reinforcement. In the case RXn=WnδXn, for some non-negative random weights W1,W2,…, the process (Xn)n≥1 is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of (Xn)n≥1 under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement.

Strong convergence of proportions in a multicolor Pólya urn

1997 ◽  
Vol 34 (2) ◽  
pp. 426-435 ◽  
Author(s):  
Raúl Gouet
Keyword(s):  
Strong Convergence ◽  
Convex Combination ◽  
Urn Model ◽  
Pólya Urn ◽  
Polya Urn ◽  
Pólya Urn Model ◽  
Generalized Pólya Urn

We prove strong convergence of the proportions Un/Tn of balls in a multitype generalized Pólya urn model, using martingale arguments. The limit is characterized as a convex combination of left dominant eigenvectors of the replacement matrix R, with random Dirichlet coefficients.

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