  An infinite urn scheme is defined by a probability mass function
$(p_j)_{j\geq1}$ over positive integers. A random allocation consists of a
sample of $N$ independent drawings according to this probability distribution
where $N$ may be deterministic or Poisson-distributed. This paper is concerned
with occupancy counts, that is with the number of symbols with $r$ or at least
$r$ occurrences in the sample, and with the missing mass that is the total
probability of all symbols that do not occur in the sample. Without any further
assumption on the sampling distribution, these random quantities are shown to
satisfy Bernstein-type concentration inequalities. The variance factors in
these concentration inequalities are shown to be tight if the sampling
distribution satisfies a regular variation property. This regular variation
property reads as follows. Let the number of symbols with probability larger
than $x$ be $\vec{\nu}(x)=|\{j:p_j\geq x\}|$. In a regularly varying urn
scheme, $\vec{\nu}$ satisfies $\lim_{\tau \rightarrow0}\vec{\nu}(\tau
x)/\vec{\nu}(\tau)=x^{-\alpha}$ for $\alpha\in[0,1]$ and the variance of the
number of distinct symbols in a sample tends to infinity as the sample size
tends to infinity. Among other applications, these concentration inequalities
allow us to derive tight confidence intervals for the Good--Turing estimator of
the missing mass.
