Dynamics of an adaptive randomly reinforced urn

Adaptive randomly reinforced urn (ARRU) is a two-color urn model where the updating process is defined by a sequence of non-negative random vectors {(D1,n, D2,n); n = 1} and randomly evolving thresholds which utilize accruing statistical information for the updates. Let m1 = E[D1,n] and m2 = E[D2,n]. In this paper, we undertake a detailed study of the dynamics of the ARRU model. First, for the case m1 = m2, we establish L1 bounds on the increments of the urn proportion, that is, the proportion of ball colors in the urn, at fixed and increasing times under very weak assumptions on the random threshold sequences. As a consequence, we deduce weak consistency of the evolving urn proportions. Second, under slightly stronger conditions, we establish the strong consistency of the urn proportions for all finite values of m1 and m2. Specifically, we show that when m1 = m2, the proportion converges to a non-degenerate random variable. Third, we establish the asymptotic distribution, after appropriate centering and scaling, for the proportion of sampled ball colors and urn proportions for the case m1 = m2. In the process, we resolve the issue concerning the asymptotic distribution of the proportion of sampled ball colors for a randomly reinforced urn (RRU). To address the technical issues, we establish results on the harmonic moments of the total number of balls in the urn at different times under very weak conditions, which is of independent interest.