scholarly journals On martingale tail sums in affine two-color urn models with multiple drawings

2017 ◽  
Vol 54 (1) ◽  
pp. 96-117 ◽  
Author(s):  
Markus Kuba ◽  
Henning Sulzbach
Keyword(s):  
Urn Model ◽  
Iterated Logarithm ◽  
Large Index ◽  
Pólya Urn ◽  
The Family ◽  
The Standard Model ◽  
Polya Urn ◽  
Random Circuits ◽  
Recursive Trees ◽  
Tail Sums

AbstractIn two recent works, Kuba and Mahmoud (2015a) and (2015b) introduced the family of two-color affine balanced Pólya urn schemes with multiple drawings. We show that, in large-index urns (urn index between ½ and 1) and triangular urns, the martingale tail sum for the number of balls of a given color admits both a Gaussian central limit theorem as well as a law of the iterated logarithm. The laws of the iterated logarithm are new, even in the standard model when only one ball is drawn from the urn in each step (except for the classical Pólya urn model). Finally, we prove that the martingale limits exhibit densities (bounded under suitable assumptions) and exponentially decaying tails. Applications are given in the context of node degrees in random linear recursive trees and random circuits.

scholarly journals Plane recursive trees, Stirling permutations and an urn model

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Svante Janson
Keyword(s):  
Joint Distribution ◽  
Urn Model ◽  
Pólya Urn ◽  
International Audience ◽  
Polya Urn ◽  
Generalized Pólya Urn ◽  
Recursive Trees

International audience We exploit a bijection between plane recursive trees and Stirling permutations; this yields the equivalence of some results previously proven separately by different methods for the two types of objects as well as some new results. We also prove results on the joint distribution of the numbers of ascents, descents and plateaux in a random Stirling permutation. The proof uses an interesting generalized Pólya urn.

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.

scholarly journals On Generalized Pólya Urn Models

2013 ◽  
Vol 50 (4) ◽  
pp. 1169-1186 ◽  
Author(s):  
May-Ru Chen ◽  
Markus Kuba
Keyword(s):  
Discrete Time ◽  
Higher Moments ◽  
Urn Models ◽  
Urn Model ◽  
Time Step ◽  
Model Case ◽  
Pólya Urn ◽  
Polya Urn ◽  
Generalized Pólya Urn

We study an urn model introduced in the paper of Chen and Wei (2005), where at each discrete time step m balls are drawn at random from the urn containing colors white and black. Balls are added to the urn according to the inspected colors, generalizing the well known Pólya-Eggenberger urn model, case m = 1. We provide exact expressions for the expectation and the variance of the number of white balls after n draws, and determine the structure of higher moments. Moreover, we discuss extensions to more than two colors. Furthermore, we introduce and discuss a new urn model where the sampling of the m balls is carried out in a step-by-step fashion, and also introduce a generalized Friedman's urn model.

Joint distributions of circular runs of various lengths based on multi-colour Pólya urn model

2012 ◽  
Vol 49 (3) ◽  
pp. 283-300
Author(s):  
Sonali Bhattacharya
Keyword(s):  
Probability Generating Function ◽  
Discrete Distribution ◽  
Time Sharing ◽  
Urn Model ◽  
Joint Distributions ◽  
Pólya Urn ◽  
Start Up ◽  
Polya Urn ◽  
Pólya Urn Model ◽  
First Time

In this paper, we have used Eryilmaz’s (2008) multi-colour Pólya urn model to obtain joint distributions of runs of t-types of exact lengths (k1, k2, …, kt), at least lengths (k1, k2, …, kt), non-overlapping runs of lengths (k1, k2, … kt) and overlapping runs of lengths (k1, k2, … kt) when counting of runs is done in a circular setup. We have also derived joint distributions of longest runs of various types under similar conditions. Distributions of runs have found applications in fields of reliability of consecutive-k-out-of n: F system, consecutive k-out-of-r-from n: F system, start-up demonstration test, molecular biology, radar detection, time sharing systems and quality control. The literature is profound in discussion of marginal distribution and joint distribution of runs of various types under linear and circular setup using techniques like urn model with balls of two or more colours, probability generating function and compounding discrete distribution with suitable beta functions. Through this paper for first time effort been made to discuss joint distributions of runs of various lengths and types using Multi-colour urn model.

A TIME-DEPENDENT PÓLYA URN WITH MULTIPLE DRAWINGS

2019 ◽  
Vol 34 (4) ◽  
pp. 469-483
Author(s):  
May-Ru Chen
Keyword(s):  
Random Variable ◽  
Time Dependent ◽  
Necessary And Sufficient Condition ◽  
Urn Model ◽  
Absolutely Continuous ◽  
Pólya Urn ◽  
Polya Urn ◽  
Generalized Pólya Urn ◽  
Necessary And Sufficient ◽  
Positive Integers

In this paper, we consider a generalized Pólya urn model with multiple drawings and time-dependent reinforcements. Suppose an urn initially contains w white and r red balls. At the nth action, m balls are drawn at random from the urn, say k white and m−k red balls, and then replaced in the urn along with cnk white and cn(m − k) red balls, where {cn} is a given sequence of positive integers. Repeat the above procedure ad infinitum. Let Xn be the proportion of the white balls in the urn after the nth action. We first show that Xn converges almost surely to a random variable X. Next, we give a necessary and sufficient condition for X to have a Bernoulli distribution with parameter w/(w + r). Finally, we prove that X is absolutely continuous if {cn} is bounded.

scholarly journals A model for the Twitter sentiment curve

PLoS ONE ◽  
2021 ◽  
Vol 16 (4) ◽  
pp. e0249634
Author(s):  
Giacomo Aletti ◽  
Irene Crimaldi ◽  
Fabio Saracco
Keyword(s):  
The Political ◽  
Data Sets ◽  
Urn Model ◽  
Public Event ◽  
Political Communications ◽  
Online Platforms ◽  
Twitter Data ◽  
Pólya Urn ◽  
Proposed Model ◽  
Polya Urn

Twitter is among the most used online platforms for the political communications, due to the concision of its messages (which is particularly suitable for political slogans) and the quick diffusion of messages. Especially when the argument stimulate the emotionality of users, the content on Twitter is shared with extreme speed and thus studying the tweet sentiment if of utmost importance to predict the evolution of the discussions and the register of the relative narratives. In this article, we present a model able to reproduce the dynamics of the sentiments of tweets related to specific topics and periods and to provide a prediction of the sentiment of the future posts based on the observed past. The model is a recent variant of the Pólya urn, introduced and studied in Aletti and Crimaldi (2019, 2020), which is characterized by a “local” reinforcement, i.e. a reinforcement mechanism mainly based on the most recent observations, and by a random persistent fluctuation of the predictive mean. In particular, this latter feature is capable of capturing the trend fluctuations in the sentiment curve. While the proposed model is extremely general and may be also employed in other contexts, it has been tested on several Twitter data sets and demonstrated greater performances compared to the standard Pólya urn model. Moreover, the different performances on different data sets highlight different emotional sensitivities respect to a public event.

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