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 $1/2$ and $1$) 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

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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