Limit Theorems for Random Triangular URN Schemes

Aguech, Rafik

doi:10.1239/jap/1253279854

Cited by 6 publications

(24 citation statements)

References 11 publications

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“…According to the above notation, we can write

, so that

. Therefore, when

, the asymptotic behaviour of

coincides with the one of

and from the results in [ 32 , 33 , 39 ] (simply translating the results proven in that papers in terms of the considered model) we immediately obtain (in the following

means almost sure convergence and

means convergence in the distribution sense): (Case

)

where D is a suitable random variable with finite moments. In particular, when

, the random variable D has probability density function given by

where c is a normalizing constant and

denotes the probability density function of the Mittag-Leffler distribution with parameter

.…”

Section: Triangular Urn Schemes and Innovation Ratesupporting

confidence: 70%

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Taylor’s Law in Innovation Processes

Tria

Crimaldi

Aletti

et al. 2020

Entropy

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Taylor’s law quantifies the scaling properties of the fluctuations of the number of innovations occurring in open systems. Urn-based modeling schemes have already proven to be effective in modeling this complex behaviour. Here, we present analytical estimations of Taylor’s law exponents in such models, by leveraging on their representation in terms of triangular urn models. We also highlight the correspondence of these models with Poisson–Dirichlet processes and demonstrate how a non-trivial Taylor’s law exponent is a kind of universal feature in systems related to human activities. We base this result on the analysis of four collections of data generated by human activity: (i) written language (from a Gutenberg corpus); (ii) an online music website (Last.fm); (iii) Twitter hashtags; (iv) an online collaborative tagging system (Del.icio.us). While Taylor’s law observed in the last two datasets agrees with the plain model predictions, we need to introduce a generalization to fully characterize the behaviour of the first two datasets, where temporal correlations are possibly more relevant. We suggest that Taylor’s law is a fundamental complement to Zipf’s and Heaps’ laws in unveiling the complex dynamical processes underlying the evolution of systems featuring innovation.

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“…According to the above notation, we can write

, so that

. Therefore, when

, the asymptotic behaviour of

coincides with the one of

and from the results in [ 32 , 33 , 39 ] (simply translating the results proven in that papers in terms of the considered model) we immediately obtain (in the following

means almost sure convergence and

means convergence in the distribution sense): (Case

)

where D is a suitable random variable with finite moments. In particular, when

, the random variable D has probability density function given by

where c is a normalizing constant and

denotes the probability density function of the Mittag-Leffler distribution with parameter

.…”

Section: Triangular Urn Schemes and Innovation Ratesupporting

confidence: 70%

“…Concerning the behavior of the number of distinct elements

, the above urn model can be seen as a triangular two-color urn scheme [ 32 , 33 , 39 , 40 ]. More precisely, we can consider an urn model with the following dynamics.…”

Section: Triangular Urn Schemes and Innovation Ratementioning

confidence: 99%

Taylor’s Law in Innovation Processes

Tria

Crimaldi

Aletti

et al. 2020

Entropy

View full text Add to dashboard Cite

show abstract

“…However, the process (B n , W n ) ∞ n=0 can be seen as a triangular urn scheme with random replacement matrix, allowing for non-negative entries. Triangular urn schemes with deterministic replacement matrix were considered by Janson [11], the results of which were extended by Aguech [1] to random triangular replacement matrices. The latter however imposed that the random variables be non-negative, to ensure the survival of the urn.…”

Section: Wn Bn+wnmentioning

confidence: 99%

“…At each time step n ≥ 1, choose a ball uniformly at random. 1. With probability p, return the ball to the urn along with another ball of the same colour.…”

Section: Introductionmentioning

confidence: 99%

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The dominating colour of an infinite Pólya urn model

Thörnblad

2016

J. Appl. Probab.

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ABSTRACT. We study a Pólya-type urn model defined as follows. Start at time 0 with a single ball of some colour. Then, at each time n ≥ 1, choose a ball from the urn uniformly at random. With probability 1/2 < p < 1, return the ball to the urn along with another ball of the same colour. With probability 1 − p, recolour the ball to a new colour and then return it to the urn. This is equivalent to the supercritical case of a random graph model studied by Backhausz and Móri [4,5] and Thörnblad [17]. We prove that, with probability 1, there is a dominating colour, in the sense that, after some random but finite time, there is a colour that always has the most number of balls. A crucial part of the proof is the analysis of an urn model with two colours, in which the observed ball is returned to the urn along with another ball of the same colour with probability p, and removed with probability 1 − p. Our results here generalise a classical result about the Pólya urn model (which corresponds to p = 1).

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Unbalanced multi-drawing urn with random addition matrix

Aguech

Selmi

2019

AJMS

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In this paper, we consider a multi-drawing urn model with random addition. At each discrete time step, we draw a sample of m balls. According to the composition of the drawn colors, we return the balls together with a random number of balls depending on two discrete random variables X and Y with finite means and variances. Via the stochastic approximation algorithm, we give limit theorems describing the asymptotic behavior of white balls.

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Limit Theorems for Random Triangular URN Schemes

Cited by 6 publications

References 11 publications

Taylor’s Law in Innovation Processes

Taylor’s Law in Innovation Processes

The dominating colour of an infinite Pólya urn model

Unbalanced multi-drawing urn with random addition matrix

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