A classical Pólya urn scheme is a Markov process whose evolution is encoded by a replacement matrix pRi,jq 1ďi,jďd . At every discrete time-step, we draw a ball uniformly at random, denote its colour c, and replace it in the urn together with Rc,j balls of colour j (for all 1 ď j ď d).We study multi-drawing Pólya urns, where the replacement rule depends on the random drawing of a set of m balls from the urn (with or without replacement). Many particular examples of this situation have been studied in the literature, but the only general results are by Kuba & Mahmoud (arXiv:1503.09069 and 1509.09053). These authors prove second order asymptotic results in the 2-colour case, under the so-called balance and affinity assumptions, the latter being somewhat artificial.The main idea of this work is to apply stochastic approximation methods to this problem, which enables us to prove analogous results to Kuba & Mahmoud, but without the artificial affinity hypothesis, and, for the first time in the literature, in the d-colour case (d ě 3). We also give some partial results in the two-colour non-balanced case, the novelty here being that the only results for this case currently in the literature are for particular examples.

We consider weighted path lengths to the extremal leaves in a random binary search tree. When linearly scaled, the weighted path length to the minimal label has Dickman's infinitely divisible distribution as a limit. By contrast, the weighted path length to the maximal label needs to be centered and scaled to converge to a standard normal variate in distribution. The exercise shows that path lengths associated with different ranks exhibit different behaviors depending on the rank. However, the majority of the ranks have a weighted path length with average behavior similar to that of the weighted path to the maximal node.

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The trie is a sort of digital tree. Ideally, to achieve balance, the trie should grow from an unbiased source generating keys of bits with equal likelihoods. In practice, the lack of bias is not always guaranteed. We investigate the distance between randomly selected pairs of nodes among the keys in a biased trie. This research complements that of Christophi and Mahmoud (2005); however, the results and some of the methodology are strikingly different. Analytical techniques are still useful for moments calculation. Both mean and variance are of polynomial order. It is demonstrated that the standardized distance approaches a normal limiting random variable. This is proved by the contraction method, whereby the limit distribution is shown to approach the fixed-point solution of a distributional equation in the Wasserstein metric space.

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