  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
$\{(D_{1,n}, D_{2,n});n\geq1\}$ and randomly evolving thresholds which utilize
accruing statistical information for the updates. Let $m_1=E[D_{1,n}]$ and
$m_2=E[D_{2,n}]$. Motivated by applications, in this paper we undertake a
detailed study of the dynamics of the ARRU model. First, for the case $m_1 \neq
m_2$, we establish $L_1$ bounds on the increments of the urn proportion at
fixed and increasing times under very weak assumptions on the random threshold
sequence. 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 $m_1$ and
$m_2$. Specifically, we show that when $m_1=m_2$ the proportion converges to a
non-degenerate random variable. Third, we establish the asymptotic
distribution, after appropriate centering and scaling, of the proportion of
sampled balls in the case $m_1=m_2$. In the process, we settle the issue of
asymptotic distribution of the number of sampled balls 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.
