  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.
