A.s. convergence for infinite colour Pólya urns associated with random walks

Janson, Svante

doi:10.48550/arxiv.1803.04207

Cited by 2 publications

(3 citation statements)

References 15 publications

(35 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Now, assume that E[ Xn ] = µ for all n, for some fixed mean µ ∈ R d . Thus, the following is a general result about random variables satisfying the distributional recursion Equation (5). The result is a special case of Theorem 4.1 and Corollary 4.2 from [14], though since the proof is short it is included for completeness.…”

Section: Contraction Methods Resultsmentioning

confidence: 94%

See 1 more Smart Citation

Random Additions in Urns of Integers

Simper¹

2019

Preprint

View full text Add to dashboard Cite

Consider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, observing the values, then replacing the balls along with a new ball labeled with the sum of the two drawn balls. This model was introduced by Siegmund and Yakir (Ann. Probab. 33(5), 2005) for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.

show abstract

Section: Contraction Methods Resultsmentioning

confidence: 94%

“…Z d ), central and local limit theorems for the random color of the n-th selected ball can be proven. This has been extended to almost sure convergence, with assumptions on the replacement scheme [5]. In all of these works, the urns are updated after drawing a single ball.…”

Section: Related Workmentioning

confidence: 99%

Random Additions in Urns of Integers

Simper¹

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…This model was introduced in [32], for specific types of weights and in full generality in [33]. It was also introduced independently by Janson in the case that all weights are one except the root, motivated by applications to infinite colour Pólya urns [34]. In [27], Sénizergues showed that a preferential attachment with additive fitness (with deterministic weights) is equal in distribution to an associated weighted random recursive tree with random weights, an interesting link between the two classes of models.…”

Section: Introductionmentioning

confidence: 99%

Degree Distributions in Recursive Trees with Fitnesses

Iyer¹

2020

Preprint

View full text Add to dashboard Cite

We study a general model of recursive trees where vertices are equipped with independent weights and at each time-step a vertex is sampled with probability proportional to its fitness function (a function of its weight and degree) and connects to ℓ new-coming vertices. Under a certain technical assumption, we derive formulas for the almost sure limiting distribution of the proportion of vertices with a given degree and weight and proportion of edges with endpoint having a certain weight. Moreover, when this assumption fails and the fitness function is affine, we show that the model can have a degenerate limiting degree distribution, or exhibit condensation where a positive proportion of edges accumulate around vertices with maximal fitness. We also prove stochastic convergence for the degree distribution under a different assumption of a strong law of large numbers for the partition function associated with the process. As an application of one of our theorems, we prove rigorously observations of Bianconi related to the evolving Cayley tree in [Phys. Rev. E 66, 036116 (2002)].

show abstract

A.s. convergence for infinite colour Pólya urns associated with random walks

Cited by 2 publications

References 15 publications

Random Additions in Urns of Integers

Random Additions in Urns of Integers

Degree Distributions in Recursive Trees with Fitnesses

Contact Info

Product

Resources

About