Azuma-Hoeffding bounds for a class of urn models

We obtain Azuma-Hoeffding bounds, which are exponentially decreasing, for the probabilities of being away from the limit for a class of urn models. The method consists of relating the variables to certain linear combinations using eigenvectors of the replacement matrix, thus bringing in appropriate martingales. Some cases of repeated eigenvalues are also considered using cyclic vectors. Moreover, strong convergence of proportions is proved as an application of these bounds.

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We present Azuma-Hoeffding bounds for a class of urn models, which show exponentially decreasing probabilities of being away from the limit. The method involves relating the variables to linear combinations using eigenvectors of the replacement matrix, and introduces appropriate martingales. Some cases of repeated eigenvalues are also considered using cyclic vectors. Moreover, the strong convergence of proportions is proved as an application of these bounds.