102 research outputs found
On martingale tail sums in affine two-color urn models with multiple drawings
In two recent works, Kuba and Mahmoud (arXiv:1503.090691 and
arXiv:1509.09053) introduced the family of two-color affine balanced Polya urn
schemes with multiple drawings. We show that, in large-index urns (urn index
between and ) 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 Polya 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.Comment: 17 page
Limit Theorems for Stochastic Approximations Algorithms With Application to General Urn Models
In the present paper we study the multidimensional stochastic approximation algorithms where the drift function h is a smooth function and where jacobian matrix is diagonalizable over C but assuming that all the eigenvalues of this matrix are in the the region Repzq ą 0. We give results on the fluctuation of the process around the stable equilibrium point of h. We extend the limit theorem of the one dimensional Robin's Monroe algorithm [MR73]. We give also application of these limit theorem for some class of urn models proving the efficiency of this method
Statistical test for an urn model with random multidrawing and random addition
We complete the study of the model introduced in [11]. It is a two-color urn
model with multiple drawing and random (non-balanced) time-dependent
reinforcement matrix. The number of sampled balls at each time-step is random.
We identify the exact rates at which the number of balls of each color grows to
infinity and define two strongly consistent estimators for the limiting
reinforcement averages. Then we prove a Central Limit Theorem, which allows to
design a statistical test for such averages.Comment: arXiv admin note: text overlap with arXiv:2102.0628
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