an:06943549
Zbl 1397.60066
Kuba, Markus; Sulzbach, Henning
On martingale tail sums in affine two-color urn models with multiple drawings
EN
J. Appl. Probab. 54, No. 1, 96-117 (2017).
00365502
2017
j
60F15 60C05 60F05 60G42
urn model; martingale central limit theorem; law of the iterated logarithm; large-index urn; triangular urn
Summary: In [``Two-colour balanced affine urn models with multiple drawings. I: Central limit theorems'', Preprint, \url{arXiv:1503.09069}; ``Two-color balanced affine urn models with multiple drawings. II: Large-index and triangular urns''', Preprint, \url{arXiv:1509.09053}], \textit{M. Kuba} and \textit{H. M. Mahmoud} introduced the family of two-color affine balanced P??lya urn schemes with multiple drawings. We show that, in large-index urns (urn index between \(\frac 12\) 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 P??lya 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.