Periodic Pólya urns and an application to Young tableaux

06/08/2018
by   Cyril Banderier, et al.
0

Pólya urns are urns where at each unit of time a ball is drawn and is replaced with some other balls according to its colour. We introduce a more general model: The replacement rule depends on the colour of the drawn ball and the value of the time (mod p). We discuss some intriguing properties of the differential operators associated to the generating functions encoding the evolution of these urns. The initial partial differential equation indeed leads to ordinary linear differential equations and we prove that the moment generating functions are D-finite. For a subclass, we exhibit a closed form for the corresponding generating functions (giving the exact state of the urns at time n). When the time goes to infinity, we show that these periodic Pólya urns follow a rich variety of behaviours: their asymptotic fluctuations are described by a family of distributions, the generalized Gamma distributions, which can also be seen as powers of Gamma distributions. En passant, we establish some enumerative links with other combinatorial objects, and we give an application for a new result on the asymptotics of Young tableaux: This approach allows us to prove that the law of the lower right corner in a triangular Young tableau follows asymptotically a product of generalized Gamma distributions.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
12/02/2019

Periodic Pólya Urns, the Density Method, and Asymptotics of Young Tableaux

Pólya urns are urns where at each unit of time a ball is drawn and repla...
research
06/12/2018

Distributions in the constant-differentials Pólya process

We study a class of unbalanced constant-differentials Pólya processes on...
research
01/13/2019

Testing for normality in any dimension based on a partial differential equation involving the moment generating function

We use a system of first-order partial differential equations that chara...
research
07/07/2021

Ergodic Numerical Approximation to Periodic Measures of Stochastic Differential Equations

In this paper, we consider numerical approximation to periodic measure o...
research
04/08/2021

Projection scheme for polynomial diffusions on the unit ball

In this article, we consider numerical schemes for polynomial diffusions...
research
11/02/2021

Asymptotic in a class of network models with sub-Gamma perturbations

For the differential privacy under the sub-Gamma noise, we derive the as...
research
08/24/2022

Recovering a probability measure from its multivariate spatial rank

We address the problem of recovering a probability measure P over ^n (e....

Please sign up or login with your details

Forgot password? Click here to reset