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

Multiple drawing multi-colour urns by stochastic approximation

Abstract A classical Pólya urn scheme is a Markov process where the evolution is encoded by a replacement matrix (Ri, j)1 ≤ i, j ≤ d. 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 ≤ j ≤ d). We study multiple 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 due to Kuba and Mahmoud (2017). These authors proved second-order asymptotic results in the two-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 and Mahmoud, but without the artificial affinity hypothesis, and, for the first time in the literature, in the d-colour case (d ≥ 3). We also provide some partial results in the two-colour nonbalanced case, the novelty here being that the only results for this case currently in the literature are for particular examples. </jats:p

Distribution of inter-node distances in digital trees

We investigate distances between pairs of nodes in digital trees (digital search trees (DST), and tries). By analytic techniques, such as the Mellin Transform and poissonization, we describe a program to determine the moments of these distances. The program is illustrated on the mean and variance. One encounters delayed Mellin transform equations, which we solve by inspection. Interestingly, the unbiased case gives a bounded variance, whereas the biased case gives a variance growing with the number of keys. It is therefore possible in the biased case to show that an appropriately normalized version of the distance converges to a limit. The complexity of moment calculation increases substantially with each higher moment; A shortcut to the limit is needed via a method that avoids the computation of all moments. Toward this end, we utilize the contraction method to show that in biased digital search trees the distribution of a suitably normalized version of the distances approaches a limit that is the fixed-point solution (in the Wasserstein space) of a distributional equation. An explicit solution to the fixed-point equation is readily demonstrated to be Gaussian

Distribution of inter-node distances in digital trees

We investigate distances between pairs of nodes in digital trees (digital search trees (DST), and tries). By analytic techniques, such as the Mellin Transform and poissonization, we describe a program to determine the moments of these distances. The program is illustrated on the mean and variance. One encounters delayed Mellin transform equations, which we solve by inspection. Interestingly, the unbiased case gives a bounded variance, whereas the biased case gives a variance growing with the number of keys. It is therefore possible in the biased case to show that an appropriately normalized version of the distance converges to a limit. The complexity of moment calculation increases substantially with each higher moment; A shortcut to the limit is needed via a method that avoids the computation of all moments. Toward this end, we utilize the contraction method to show that in biased digital search trees the distribution of a suitably normalized version of the distances approaches a limit that is the fixed-point solution (in the Wasserstein space) of a distributional equation. An explicit solution to the fixed-point equation is readily demonstrated to be Gaussian