scholarly journals The Number of Two Consecutive Successes in a Hoppe-Pólya Urn

2008 ◽  
Vol 45 (3) ◽  
pp. 901-906 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Explicit Formula ◽  
Bernoulli Trials ◽  
Pólya Urn ◽  
Probability Of Success ◽  
Polya Urn ◽  
Binomial Moments

In a sequence of independent Bernoulli trials the probability of success in the kth trial is pk = a / (a + b + k − 1). An explicit formula for the binomial moments of the number of two consecutive successes in the first n trials is obtained and some consequences of it are derived.

scholarly journals The Number of Two Consecutive Successes in a Hoppe-Pólya Urn

2008 ◽  
Vol 45 (03) ◽  
pp. 901-906 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Explicit Formula ◽  
Bernoulli Trials ◽  
Pólya Urn ◽  
Probability Of Success ◽  
Polya Urn ◽  
Binomial Moments

In a sequence of independent Bernoulli trials the probability of success in the kth trial is p k = a / (a + b + k − 1). An explicit formula for the binomial moments of the number of two consecutive successes in the first n trials is obtained and some consequences of it are derived.

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

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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